Propagation of double zero-order Bessel beam and its scattering properties to uniaxial anisotropic spheres

Li Shun; Li Zheng-Jun; Qu Tan; Li Hai-Ying; Wu Zhen-Sen

doi:10.7498/aps.71.20220491

Abstract

Authors and contacts

FullText HTML

1. 引　言
2. 散射理论
2.1 零阶贝塞尔波束的描述
2.2 双零阶贝塞尔波束的展开
2.3 双零阶贝塞尔波束的传播特性
2.4 UA介质球的内场展开
2.5 通过边界条件求解散系数
3. 数值结果和讨论
4. 结　论

References

Cited By

Abstract

Based on the generalized Lorenz Mie theory, the propagation and scattering properties of a uniaxial anisotropic spherical particle illuminated separately by double zero-order Bessel beam with arbitrary propagation direction and polarization direction are studied. The propagation and scattering characteristics are compared with those of a uniaxial anisotropic spherical particle illuminated by a single zero-order Bessel beam. Using the orthogonal relation of the spherical vector wave function and coordinate rotation theorem, the expanded forms of double zero-order Bessel beams with arbitrary propagation direction and polarization direction are derived. The analytical expressions of the expansion coefficients are derived by the integral method. The expansion coefficients of total incident field are obtained through the vector superposition principle. Based on the Fourier transform and tangentially continuous boundary conditions, the internal electromagnetic field of the uniaxial anisotropic sphere is expanded in terms of the spherical vector wave function and the scattering coefficients are derived. By comparing the angular distribution of the radar cross section of the particle illuminated by single and double zero-order Bessel beam when degenerating into plane waves with those results given by the literature, the correctness of the theory and the program in this paper are both verified. The effects of the incidence angle, conic angle and polarization angle on angle distribution of the radar cross section are numerically analyzed. The theoretical and numerical results in this paper are expected to be used to study the scattering properties, particle size analysis and optical trapping for anisotropic particles, biological cells and other particles illuminated by multi-beams.

Keywords:

PACS:

03.65.Nk	(Scattering theory)
42.55.Ah	(General laser theory)
42.68.Mj	(Scattering, polarization)

Authors and contacts

1.
School of Physics, Xidian University, Xi’an 710071, China
2.
Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China
3.
School of Electronic Engineering, Xidian University, Xi’an 710071, China

Corresponding author: Li Zheng-Jun, lizj@xidian.edu.cn

Funds: Project supported by the Natural Science Basic Research Program of Shaanxi Province, China (Grant No. 2021JM-135) and the National Natural Science Foundation of China (Grant Nos. 62071359, 61801349).

FullText HTML

1. 引　言

自从Durnin^[1]提出贝塞尔波束以来, 由于其非衍射和自重构特性, 受到越来越多的关注, 并广泛应用于不同的领域, 如光捕获和操作、粒径分析、光导和对准等^[2-5]. 作为激光束, 许多学者对贝塞尔波束在球面和柱面坐标系中的轮廓和描述进行了深入研究^[6-11]. 基于该描述, 许多学者进一步研究了球形粒子对贝塞尔波束的散射问题. Marson^[12]使用平面波分解方法研究了球体对零阶贝塞尔的散射特性, Ma和Li^[13]研究了球形粒子对非偏振贝塞尔波束的散射. 入射贝塞尔波束被认为是Cimár等^[8]提出的平面波的叠加, 并在球面矢量波函数方面进行了展开, 实际上它是轴棱镜生成的贝塞尔波束. 基于广义洛伦兹Mie理论^[14], Li等^[15]研究了由棱镜生成的贝塞尔波束照射的球形粒子的散射. 用积分局域近似法(integral local approximation, ILA), Ambrosio和Herández-Figueroa^[10]给出了零阶贝塞尔波束的球矢量波函数展开系数. 然而, 贝塞尔波束由标量波理论描述, 只有当圆锥角比较小时才能提供令人满意的结果. Mishra^[11]则利用矢量波理论来描述任意圆锥角的贝塞尔波束, 得到了比较满意无衍射特性的贝塞尔波束. 使用这种矢量描述, Mitri^[16]研究了各向同性球形粒子对零阶贝塞尔波束的散射特性. 利用数值方法, Cui等^[17]研究了任意形状的均匀介电粒子对零阶贝塞尔波束的散射特性. Klimov^[18]导出了零阶贝塞尔波束在球体上散射的解析解, Wang等^[19]研究了零阶贝塞尔波束激发的受激拉曼散射显微成像用于散射组织中深层目标的成像. 但是上述研究主要针对简单的各向同向粒子的散射问题, 关于各向异性粒子对零阶贝塞尔波束散射特性的研究还比较少.

近年来, 各向异性介质的散射特性因其在光信号处理、雷达散射截面(radar cross section, RCS)控制和微波器件制造等领域的应用越来越广泛而备受关注. Stout等^[20]使用微分理论给出了由任意各向异性介质构成的任意形状物体的散射解. Wong等^[21]和Qiu等^[22]研究了单轴各向异性(uniaxial anisotropy, UA)球形粒子对平面波的散射特性. 通过引入傅里叶变换的方法, Geng等^[23]利用解析方法研究了UA球形粒子对平面波的散射特性, Wang等^[24]研究了旋转各向异性介质球对离轴高斯波束的散射特性. 近十年, Yuan等^[25,26]深入研究了在轴、离轴和任意入射高斯波束照射下UA球体的散射特性. Wang等^[27]基于T矩阵方法研究了UA介质球对高斯波束的散射特性, 并与解析方法进行了比较验证. Qu等^[28]研究了UA球体对离轴零阶贝塞尔波束的散射特性.

然而这些研究中, 入射波仅限于平面波或单个贝塞尔波束, 其中各向异性球体对双零阶贝塞尔波束散射特性的研究还是非常少的. 特别地, 由于主光轴的存在, 零阶贝塞尔波束离轴斜入射到UA粒子上时, 其散射特性与在轴和离轴入射情况下UA粒子的散射特性有很大不同. 并且, 与单波束光势阱相比, 双波束光势阱或驻波光势阱用于非常小的同位素球形粒子及非各向同性粒子的光学捕获更加具有优势^[29]. 这就要求对于非各向同性球形粒子对双聚焦波束的散射理论及特性问题要研究透彻. 基于此, 本文主要研究了UA球形粒子对于双任意方向入射零阶贝塞尔波束的散射问题, 并详细讨论了双零阶贝塞尔波束的电场强度分布及作用于UA球形粒子RCS角分布的情况. 值得注意的是, 本文的研究与广义洛伦兹Mie理论不同的地方主要有两点: 一是入射波束从高斯波束入射扩展到任意方向传播双零阶贝塞尔波束入射, 其球矢量波函数的展开利用了坐标旋转定理, 展开系数是通过积分方法及矢量加法定理得到的; 二是内场的展开要比各向同性球形粒子要复杂得多, 基于傅里叶变换方法根据球矢量波函数L, M和N的正交完备性进行展开. 因而, UA球体对于双任意方向入射零阶贝塞尔波束散射问题的研究是广义洛伦兹Mie理论的深入的扩展.

2. 散射理论

2.1 零阶贝塞尔波束的描述

作为一种典型的有形波束, 零阶贝塞尔波束的标量形式可以看作是标量波动方程的一类精确解. 假设时谐因子为 ${{\text{e}}^{ - {\text{i}}\omega t}}$ ( $\omega$ 为角频率), 零阶贝塞尔波束电场在波束坐标系 $Oxyz$ 中可表示为^[1]

$E = {E_0}{{\text{J}}_0}\left( {{k_R}R} \right){{\text{e}}^{{\text{i}}{k_z}z}}{{\text{e}}^{ - {\text{i}}\omega t}} \text{, }$

(1)

其中 ${E_0}$ 为归一化振幅, ${{\text{J}}_0}\left( \cdot \right)$ 为零阶柱阶贝塞尔函数, 参数 $R = \sqrt {{x^2} + {y^2}}$ 以及 $\phi = {\tan ^{ - 1}}\left( {y/x} \right)$ 分别为横向面平面 $\left( {x, y} \right)$ 上的半径和方位角, ${k_R} = {k_0}\sin {C_0}$ 和 ${k_z} = {k_0}\cos {C_0}$ 分别为波矢k的横向分量和纵向分量, ${C_0}$ 为波束的半锥角.

一般来说, 当半锥角小于10°时, 上述标量表示(1)式足以精确地描述零阶贝塞尔波束的无衍射特性. 但是, 当半锥角比较大时: 例如 ${k_R} \approx {k_0}$ , (1)式所表示的贝塞尔波束就会与实际情况出现比较大的误差. 因此, Mishra^[11]通过两个矢势来构建一个电磁场, 得到具有对称性的电场强度和磁场强度的表达式. 矢量描述是通过引入矢量势 ${\boldsymbol{A}}$ 来定义零阶贝塞尔波束的电磁场^[30]:

$\begin{split} &{\boldsymbol{H}}\sqrt {{\mu _0}/{\varepsilon _0}} = \nabla \times {\boldsymbol{A}}{\text{ ,}} \\&{\boldsymbol{E}} = {\text{i}}{k_0}[{\boldsymbol{A}} + \nabla (\nabla \cdot {\boldsymbol{A}})/k_0^2]{\text{ }}{\text{. }} \\ \end{split}$

(2)

假设矢势 ${\boldsymbol{A}}$ 沿x轴方向偏振, 通过求解亥姆霍兹方程, 可以得到

${\boldsymbol{A}} = {{\boldsymbol{\hat {e}}}_x}{A_0}{{\text{J}}_0}({k_R}R)\exp ({\text{i}}{k_z}z) .$

(3)

对于矢量波方法, 为得到圆对称的零阶贝塞尔波束, 设沿y方向偏振的矢势 ${\boldsymbol{A}}'$ , 则

${\boldsymbol{A}}' = - {{\boldsymbol{\hat {e}}}_y}{A_0}{{\text{J}}_0}({k_R}R)\exp ({\text{i}}{k_z}z) .$

(4)

${\boldsymbol{E}}'$ 和 ${\boldsymbol{H}}'$ 的表达式为

$\begin{split} \begin{split}{{\boldsymbol{E}}'}=\;&\nabla \times {{\boldsymbol{A}}'}\text{, }\\ {{\boldsymbol{H}}'}=\;&{\rm{i}}k[{{\boldsymbol{A}}'}+\nabla (\nabla \cdot {{\boldsymbol{A}}'})]. \end{split} \end{split}$

(5)

(5)式与(2)式结果相加再取平均, 即可得到沿z轴传播、x轴偏振的零阶贝塞尔波束在其波束坐标系 $Oxyz$ 中的电磁场表示式^[11,16]:

$\begin{split} {\boldsymbol{E}}=\;&\frac{1}{2}{E}_{0}{\text{e}}^{\text{i}{k}_{z}z}\left\{ \left [\left(1+\frac{{k}_{z}}{{k}_{0}}-\frac{{k}_{R}^{2}{x}^{2}}{{k}_{0}^{2}{R}^{2}} \right){\text{J}}_{\text{0}}({k}_{R}R)\right.\right. \left.-\;{\text{J}}_{\text{1}}\left({k}_{R}R\right)\frac{{k}_{R}\left({y}^{2}-{x}^{2}\right)}{{k}_{0}^{2}{R}^{3}} \right]{{\boldsymbol{\hat{e}}}}_{x}\\ &+xy\left[{\text{J}}_{\text{1}}\left({k}_{R}R\right)\frac{2{k}_{R}}{{k}_{0}^{2}{R}^{3}}-\frac{{k}_{R}^{2}}{{k}_{0}^{2}{R}^{2}}{\text{J}}_{\text{0}}({k}_{R}R)\right]{{\boldsymbol{\hat{e}}}}_{y} \left.+\frac{x}{\text{i}{k}_{0}R}\left(1+\frac{{k}_{z}}{{k}_{0}}\right){k}_{R}{\text{J}}_{\text{1}}({k}_{R}R){{\boldsymbol{\hat{e}}}}_{z}\right\}\text{, }\\[-8pt]\end{split}$

(6)

$\begin{split}{\boldsymbol{H}}=\;&\frac{1}{2}{E}_{0}\sqrt{{\varepsilon }_{0}/{\mu }_{0}}{\text{e}}^{\text{i}{k}_{z}z}\left\{ xy\left[\frac{2{k}_{R}}{{k}_{0}^{2}{R}^{3}}{\text{J}}_{\text{1}}({k}_{R}R)\right. -\;\frac{{k}_{R}^{2}}{{k}_{0}^{2}{R}^{2}}{\text{J}}_{\text{0}}({k}_{R}R)\right]{{\boldsymbol{\hat{e}}}}_{x}-\left[{\text{J}}_{\text{1}}({k}_{R}R){k}_{R}\frac{{x}^{2}-{y}^{2}}{{k}_{0}^{2}{R}^{3}}\right.\\ &\left.+\left(1+\frac{{k}_{z}}{{k}_{0}}-\frac{{y}^{2}}{{k}_{0}^{2}{R}^{2}}{k}_{R}^{2}\right){\text{J}}_{\text{0}}({k}_{R}R)\right]{{\boldsymbol{\hat{e}}}}_{y} \left.+\frac{{k}_{R}y}{\text{i}kR}\left(1+\frac{{k}_{z}}{{k}_{0}}\right){\text{J}}_{\text{1}}\left({k}_{R}R\right){{\boldsymbol{\hat{e}}}}_{z}\right\}\text{, }\\[-15pt]\end{split}$

(7)

其中 ${{\boldsymbol{\hat e}}_x}$ , ${{\boldsymbol{\hat e}}_y}$ 和 ${{\boldsymbol{\hat e}}_z}$ 分别为 $Oxyz$ 中的单位方向矢量.

2.2 双零阶贝塞尔波束的展开

如图1(a)所示, 两束零阶贝塞尔波束离轴斜入射到UA介质球上, 两波束中心分别为 ${O_1}$ 和 ${O_2}$ . ${O_1}{x_1}{y_1}{z_1}$ 和 ${O_2}{x_2}{y_2}{z_2}$ 分别是以各自波束中心为原点, 传播方向为 ${z_1}$ 和 ${z_2}$ 轴建立的直角坐标系. $Oxyz$ 是以UA介质球心 $O$ 为原点建立的直角坐标系, 设定各向异性介质球的主光轴沿 $z$ 轴方向, 半径为 $a$ . 设定在粒子坐标系 $Oxyz$ 中, 两波束中心 ${O_1}$ 和 ${O_2}$ 的坐标分别为 $({x_1}, {y_1}, {z_1})$ 和 $({x_2}, {y_2}, {z_2})$ , 两束零阶贝塞尔波束的半锥角分别用 ${C_1}$ 和 ${C_2}$ 表示, 传播方向分别用 ${k_1}$ 和 ${k_2}$ 表示. 如图1(b)所示, 设定 ${\alpha _1}$ 为第一束零阶贝塞尔波束的入射角, 是波束传播方向 ${k_1}$ 与粒子坐标系 $Oxyz$ 中 $z$ 轴的夹角; ${\beta _1}$ 为第一束零阶贝塞尔波束的极化角, 是波束传播方向 ${k_1}$ 在粒子坐标系 $Oxyz$ 下 $xOy$ 面上的投影与 $x$ 轴的夹角, 这样可以表示出零阶贝塞尔波束的传播和极化方向的任意性. 类似地, 可以设定 ${\alpha _2}$ 和 ${\beta _2}$ 为第二束零阶贝塞尔波束的入射角和极化角. 这里与图1(b)的情况相同, 只是角度的表示不一样, 不再重复画出示意图. 值得注意的是, 由于UA介质存在主光轴, 当波束的入射方向与主光轴不同时会产生不同的散射特性. 入射角定义为0°—180°的范围, 相对各向异性球形粒子的主光轴 $z$ 轴而言, 零阶贝塞尔波束是以任意方向入射UA介质球的.

$图 1 双零阶贝塞尔波束离轴斜入射UA介质球的示意图\r\nFig. 1. Schematic diagram of a uniaxial anisotropic sphere illuminated by off-axis obliquely incident double zero-order Bessel beams.$

图 1 双零阶贝塞尔波束离轴斜入射UA介质球的示意图

Fig. 1. Schematic diagram of a uniaxial anisotropic sphere illuminated by off-axis obliquely incident double zero-order Bessel beams.

下载: 全尺寸图片幻灯片

将(6)式和(7)式中相应的坐标表示 $(x, y, z)$ 分别替代成坐标表示 $({x_1}, {y_1}, {z_1})$ 和 $({x_2}, {y_2}, {z_2})$ , 即可以得到两束零阶贝塞尔波束在各自坐标系 ${O_1}{x_1}{y_1}{z_1}$ 和 ${O_2}{x_2}{y_2}{z_2}$ 的电磁场矢量表示式. 由于零阶贝塞尔波束的矢量表示式本身就是从麦克斯韦方程组推导得来的, 因此可以将这两束零阶贝塞尔波束根据球矢量波函数进行展开. 以第一束零阶贝塞尔波束为例, 可根据球矢量波函数在粒子坐标系 $Oxyz$ 下展开:

$\begin{split}{{\boldsymbol{E}}}^{\text{i1}}\left({r}^{1}\right)=\;&\displaystyle \sum _{n=1}^{\infty }\displaystyle \sum _{m=-n}^{n}\left[{a}_{mn}^{\text{i1}}{{\boldsymbol{M}}}_{mn}^{(1)}\left({r}^{1},{k}^{1}\right)\right.\\& \left.+{b}_{mn}^{\text{i1}}{{\boldsymbol{N}}}_{mn}^{(1)}\left({r}^{1},{k}^{1}\right)\right]\text{, }\\ {{\boldsymbol{H}}}^{\text{i1}}\left({r}^{1}\right)=\;&\frac{{k}_{0}}{\text{i}\omega {\mu }_{0}}\displaystyle \sum _{n=1}^{\infty }\displaystyle \sum _{m=-n}^{n}\left[{a}_{mn}^{\text{i1}}{{\boldsymbol{M}}}_{mn}^{(1)}\left({r}^{1},{k}^{1}\right)\right.\\& \left.+{b}_{mn}^{\text{i1}}{{\boldsymbol{N}}}_{mn}^{(1)}\left({r}^{1},{k}^{1}\right)\right]\text{, }\\[-5pt]\end{split}$

(8)

其中 $a_{mn}^{{\text{i}}1}$ 和 $b_{mn}^{{\text{i}}1}$ 表示展开系数, 上标1表示与第一束贝塞尔波束相关的量. ${\boldsymbol{M}}_{mn}^{(1)}\left( {r, k} \right)$ 和 ${\boldsymbol{N}}_{mn}^{(1)}\left( {r, k} \right)$ 代表球矢量波函数, 其具体表达式为^[31]

$\begin{split} &{\boldsymbol{M}}_{mn}^{(l)}\left( {r,k} \right)= \\ \;&z_n^{(l)} ({kr}) \left[ {{\text{i}}m\frac{{{\rm{P}}_n^m (\cos \theta)}}{{\sin \theta }}{{\text{e}}^{{\text{i}}m\phi }}{{{\boldsymbol{\hat e}}}_\theta } - \frac{{{\text{d}}{\rm{P}}_n^m}}{{{\text{d}}\theta }}{{\text{e}}^{{\text{i}}m\phi }}{{{\boldsymbol{\hat e}}}_\phi }} \right] \text{, } \end{split}$

(9)

$\begin{split}&{{\boldsymbol{N}}}_{mn}^{(l)}\left(r,k\right)\\ =\;&n\left(n + 1\right)\frac{{z}_{n}^{\left(l\right)}\left(kr\right)}{kr}{{\rm{P}}}_{n}^{m}\left(\mathrm{cos}\theta \right){\text{e}}^{\text{i}m\phi }{{\boldsymbol{\hat{e}}}}_{r}\text{+}\frac{1}{kr}\frac{\text{d} \left( r{z}_{n}^{\left(l\right)} \left(kr\right) \right)}{\text{d}r}\\ & \times\left[\frac{\text{d}{{\rm{P}}}_{n}^{m}\left(\mathrm{cos}\theta \right)}{\text{d}\theta }{{\boldsymbol{\hat{e}}}}_{\theta }+\text{i}m\frac{{{\rm{P}}}_{n}^{m}\left(\mathrm{cos}\theta \right)}{\mathrm{sin}\theta }{{\boldsymbol{\hat{e}}}}_{\phi }\right]{\text{e}}^{\text{i}m\phi }. \\[-15pt]\end{split}$

(10)

当 $l = 1, {\text{ }}2, {\text{ }}3, {\text{ }}4$ 时, $z_n^{\left( l \right)}\left( {kr} \right)$ 分别是球贝塞尔函数 ${{\text{j}}_n}\left( {kr} \right)$ 、球诺依曼函数 ${{\text{y}}_n}\left( {kr} \right)$ 、第一类和第二类球Hankel函数 ${\rm{h}}_n^{\left( 1 \right)}\left( {kr} \right)$ 和 ${\rm{h}}_n^{\left( 2 \right)}(kr)$ . 根据连带勒让德函数 ${\rm{P}}_n^m\left( {\cos \theta } \right)$ 和指数函数的正交关系, 可将上述入射场展开系数 $a_{mn}^{{\text{i}}1}$ 和 $b_{mn}^{{\text{i}}1}$ 用积分方法表示为^[32,33]

$\begin{split} \;& \left( \begin{gathered} a_{mn}^{{\text{i}}1} \\ b_{mn}^{{\text{i}}1} \end{gathered} \right) = \left( \begin{gathered} \sqrt{\dfrac{\mu_0}{\varepsilon _0}} \\ 1 \end{gathered} \right) \frac{{(2n + 1){{({k_0}a)}^2}}}{{{E_0}4\pi n(n + 1){\psi _n}({k_0}a)}} \frac{{( {n - m} )!}}{{( {n + m} )!}} \\ & \times \int_0^{2{\text{π}}} \int_0^{\text{π}} \left(\begin{array}{*{20}{c}} {\text{i}}{\boldsymbol{H}}_{{r_1} = a}^i \\ {\boldsymbol{E}}_{{r_1} = a}^i \end{array} \right) {\rm{P}}_n^m\left( {\cos \theta } \right){{\text{e}}^{ - {\text{i}}m\phi }}\sin \theta {\text{d}}\theta {\text{d}}\phi \text{, } \end{split}$

(11)

其中, ${\varepsilon _0}$ 和 ${\mu _0}$ 分别为在真空传播时的介电常数和磁导率, ${{\psi}_n}({k_0}a) = {k_0}a{{\text{j}}_n}({k_0}a)$ 为黎卡提贝塞尔函数. 如果已知原始零阶贝塞尔波束在粒子坐标系 $Oxyz$ 下的径向电磁场, 则可以通过求解方程(11)式中的二重积分来得到展开系数. 需要注意的是, 零阶贝塞尔波束的入射方向不平行于z轴, 因此无法通过简单的坐标平移得到波束的径向电磁场. 可以通过坐标系的旋转理论来解决这个问题. 如图2所示, 以第一束零阶贝塞尔波束中心 ${O_1}$ 为原点建立与粒子坐标系 $Oxyz$ 相互平行的中间直角坐标系 ${O_1}{x_{10}}{y_{10}}{z_{10}}$ . 一般地, 任意两个非平行的直角坐标系都可以通过欧勒角 $\alpha$ , $\beta$ , $\gamma$ 角相互旋转得到^[34,35]. 为了简化, 这里只用两个角即 ${\alpha _1}$ , ${\beta _1}$ . 首先将直角坐标系 ${O_1}{x_1}{y_1}{z_1}$ 绕 ${y_1}$ 轴逆时针旋转 ${\alpha _1}$ 角, 得到 ${z_1}$ 与 ${z_{10}}$ 轴重合; 再将 ${O_1}{x_1}{y_1}{z_1}$ 以 ${z_1}$ 为轴逆时针旋转 ${\beta _1}$ 角, 即可以使直角坐标系 ${O_1}{x_1}{y_1}{z_1}$ 与 ${O_1}{x_{10}}{y_{10}}{z_{10}}$ 相互重合.

$图 2 两直角坐标的旋转关系示意图\r\nFig. 2. Schematic diagram of rotation relationship between two rectangular coordinate systems.$

图 2 两直角坐标的旋转关系示意图

Fig. 2. Schematic diagram of rotation relationship between two rectangular coordinate systems.

下载: 全尺寸图片幻灯片

通过图2的旋转关系, 可以得到两个坐标系下坐标的关系:

$\begin{split} & \left[ \begin{gathered} {x_{10}} \\ {y_{10}} \\ {z_{10}} \end{gathered} \right] = {\boldsymbol{A}}\left[ \begin{gathered} {x_1} \\ {y_1} \\ {z_1} \end{gathered} \right] =\\ & \left( {\begin{array}{*{20}{c}} {\cos {\alpha _1}\cos {\beta _1}}&{ - \sin {\beta _1}}&{\sin {\alpha _1}\cos {\beta _1}} \\ {\cos {\alpha _1}\sin {\beta _1}}&{\cos {\beta _1}}&{\sin {\alpha _1}\sin {\beta _1}} \\ { - \sin {\alpha _1}}&0&{\cos {\alpha _1}} \end{array}} \right)\left[ \begin{gathered} {x_1} \\ {y_1} \\ {z_1} \end{gathered} \right] . \end{split}$

(12)

基于上述旋转关系, 第一束零阶贝塞尔波束在波束坐标系 ${O_1}{x_1}{y_1}{z_1}$ 下的电磁场分量 ${E_{{x_1}}}$ , ${E_{{y_1}}}$ , ${E_{{z_1}}}$ 和 ${H_{{x_1}}}$ , ${H_{{y_1}}}$ , ${H_{{z_1}}}$ 可转换到中间坐标下 ${O_1}{x_{10}}{y_{10}}{z_{10}}$ 下的场分量 ${E_{{x_{10}}}}$ , ${E_{{y_{10}}}}$ , ${E_{{z_{10}}}}$ 和 ${H_{{x_{_{10}}}}}$ , ${H_{{y_{10}}}}$ , ${H_{{z_{10}}}}$ :

$\begin{split} \left[ \begin{gathered} {E_{{x_{10}}}} \\ {E_{{y_{10}}}} \\ {E_{{z_{10}}}} \end{gathered} \right] = {\boldsymbol{A}}\left[ \begin{gathered} {E_{{x_1}}} \\ {E_{{y_1}}} \\ {E_{{z_1}}} \end{gathered} \right],\;\; \left[ \begin{gathered} {H_{{x_{10}}}} \\ {H_{{y_{10}}}} \\ {H_{{z_{10}}}} \end{gathered} \right] = {\boldsymbol{A}}\left[ \begin{gathered} {H_{{x_1}}} \\ {H_{{y_1}}} \\ {H_{{z_1}}} \end{gathered} \right] . \end{split}$

(13)

由于中间坐标系 ${O_1}{x_{10}}{y_{10}}{z_{10}}$ 与粒子坐标系 $Oxyz$ 的坐标之间有如下关系:

$\left[ \begin{gathered} {x_{10}} + {x_1} \\ {y_{10}} + {y_1} \\ {z_{10}} + {z_1} \end{gathered} \right] = \left[ \begin{gathered} x \\ y \\ z \end{gathered} \right] \text{, }$

(14)

因此可以得到第一束零阶贝塞尔波束在粒子坐标系下的电磁场径向分量, 再代入(11)式即可以获得第一束零阶贝塞尔波束在粒子坐标系 $Oxyz$ 下的展开系数.

类似地, 建立一个平行于粒子坐标系 $Oxyz$ 的中间坐标系 ${O_2}{x_{20}}{y_{20}}{z_{20}}$ , 第二束零阶贝塞尔波束也可以根据球矢量波函数在粒子坐标系 $Oxyz$ 下展开:

$\begin{split}{{\boldsymbol{E}}}^{\text{i2}}=\;&\displaystyle \sum _{n=1}^{\infty }\displaystyle \sum _{m=-n}^{n}\left[{a}_{mn}^{\text{i2}}{{\boldsymbol{M}}}_{mn}^{(1)}({r}^{20},{k}^{2})\right.\\& \left.+\;{b}_{mn}^{\text{i1}}{{\boldsymbol{N}}}_{mn}^{(1)}({r}^{20},{k}^{2})\right]\text{, }\\ {{\boldsymbol{H}}}^{\text{i2}}=\;&\frac{{k}^{2}}{\text{i}{\omega }_{1}{\mu }_{0}}\displaystyle \sum _{n=1}^{\infty }\displaystyle \sum _{m=-n}^{n}\left[{a}_{mn}^{\text{i2}}{{\boldsymbol{N}}}_{mn}^{(1)}({r}^{20},{k}^{2})\right.\\& \left.+\;{b}_{mn}^{\text{i2}}{{\boldsymbol{M}}}_{mn}^{(1)}({r}^{20},{k}^{2})\right]. \\[-6pt]\end{split}$

(15)

同样可以得到第二束零阶贝塞尔波束在粒子坐标系下的展开系数:

$\begin{split} \;& \left( \begin{gathered} a_{mn}^{{\text{i}}2} \\ b_{mn}^{{\text{i}}2} \end{gathered} \right) = \left( \begin{gathered} \sqrt{\dfrac{\mu _0}{\varepsilon _0}} \\ 1 \end{gathered} \right) \frac{{(2n + 1){{({k_0}a)}^2}}}{{{E_0}4\pi n(n + 1){\psi _n}({k_0}a)}} \frac{{( {n - m} )!}}{{( {n + m} )!}} \\ & \times \int_0^{2{\text{π}}} \int_0^{\text{π}} \left( \begin{array}{*{20}{c}} {\text{i}}{\boldsymbol{H}}_r^{{\text{i}}2} \\ {\boldsymbol{E}}_r^{{\text{i}}2} \end{array} \right)_{r = a}{\rm{P}}_n^m\left( {\cos \theta } \right){{\text{e}}^{ - {\text{i}}m\phi }}\sin \theta {\text{d}}\theta {\text{d}}\phi. \\[-8pt] \end{split}$

(16)

对于两束入射的零阶贝塞尔波束, 可通过两个电磁场的矢量叠加来计算总入射场, 则总入射场展开系数为

${a}_{mn}^{{\rm{it}}}={a}_{mn}^{\text{i}1}+{a}_{mn}^{\text{i2}},\;\;{b}_{mn}^{{\rm{it}}}={b}_{mn}^{\text{i}1}+{b}_{mn}^{\text{i2}}.$

(17)

2.3 双零阶贝塞尔波束的传播特性

图3为不同半锥角下沿z轴正、负方向相同极化角时, 反向传播的双零阶贝塞尔波束在 $xOy$ 面的电场强度分布情况. 零阶贝塞尔波束电场振幅 ${E_0}$ 设定为1, 计算参数为: ${\alpha _1} = {\beta _1} = {\beta _2} = 0^\circ$ , ${\alpha _2} = 180^\circ$ , ${z_1} = 1\text{ }{{ \text{μm}}}$ , ${z_2} = - 1\text{ }{{ \text{μm}}}$ (后文若无特殊说明, E₀ = 1, z₁与z₂保持不变). 与单束零阶贝塞尔波束相比, 反向传播双零阶贝塞尔波束在 $xOy$ 面的电场强度分布形成了4个明显的孤岛. 同时, 半锥角越大, 反向传播双零阶贝塞尔波束在 $xOy$ 的场值整体越强, 4个孤岛更加向中心区域集中, 周边区域的电场强度分布也会形成更多的孤岛.

$图 3 相同极化角下反向传播双零阶贝塞尔波束的电场强度分布　(a) ${C_1} = {C_2} = 5^\circ $; (b) ${C_1} = {C_2} = 15^\circ $; (c) ${C_1} = {C_2} = 30^\circ $\r\nFig. 3. Electric intensity distribution of back propagating double zero-order Bessel beams with identical polarization angles: (a) ${C_1} = $$ {C_2} = 5^\circ $; (b) ${C_1} = {C_2} = 15^\circ $; (c) ${C_1} = {C_2} = 30^\circ $.$

图 3 相同极化角下反向传播双零阶贝塞尔波束的电场强度分布　(a)

${C_1} = {C_2} = 5^\circ$

; (b)

${C_1} = {C_2} = 15^\circ$

; (c)

${C_1} = {C_2} = 30^\circ$

Fig. 3. Electric intensity distribution of back propagating double zero-order Bessel beams with identical polarization angles: (a)

${C_1} =$

${C_2} = 5^\circ$

; (b)

${C_1} = {C_2} = 15^\circ$

; (c)

${C_1} = {C_2} = 30^\circ$

下载: 全尺寸图片幻灯片

如图4所示, 计算了不同半锥角下沿z轴正、负方向不同极化角时, 反向传播的双零阶贝塞尔波束在 $xOy$ 面的电场强度分布情况. 计算参数为: ${\alpha _1}={\beta _1}=0^\circ$ , ${\alpha _2} = {\beta _2}= 180^\circ$ . 与极化角相同时反向传播双零阶贝塞尔波束(图3)相比, 形成的孤岛有所减少, 而孤岛中心强度却明显增加, 形成孤岛的位置和方位也发生了较大的改变. 在与粒子相互作用中, 两束零阶贝塞尔波束极化角不同时, 散射特性应该会不同.

$图 4 不同极化角下反向传播双零阶贝塞尔波束的电场强度分布　(a) ${C_1} = {C_2} = 5^\circ $; (b) ${C_1} = {C_2} = 15^\circ $; (c) ${C_1} = {C_2} = 30^\circ $\r\nFig. 4. Electric intensity distribution of back propagating double zero-order Bessel beams with different polarization angles: (a) ${C_1} = {C_2} = 5^\circ $; (b) ${C_1} = {C_2} = 15^\circ $; (c) ${C_1} = {C_2} = 30^\circ $.$

图 4 不同极化角下反向传播双零阶贝塞尔波束的电场强度分布　(a)

${C_1} = {C_2} = 5^\circ$

; (b)

${C_1} = {C_2} = 15^\circ$

; (c)

${C_1} = {C_2} = 30^\circ$

Fig. 4. Electric intensity distribution of back propagating double zero-order Bessel beams with different polarization angles: (a)

${C_1} = {C_2} = 5^\circ$

; (b)

${C_1} = {C_2} = 15^\circ$

; (c)

${C_1} = {C_2} = 30^\circ$

下载: 全尺寸图片幻灯片

图5给出了不同半锥角下斜入射反向传播的双零阶贝塞尔波束在 $xOy$ 面的电场强度分布情况. 计算参数为: ${\alpha _1} = 30^\circ$ , ${\alpha _2} = 210^\circ$ , ${\beta _1}={\beta _2}=0^\circ$ . 虽然双零阶贝塞尔波束的传播方向是反向的, 但是电场强度分布并没有形成孤岛. 因为斜入射情况下, 双波束相向传播到波束中心时, 应该在与其传播方向相互垂直的面上形成孤岛而不是 $xOy$ 面上. 随着半锥角的增大, 双零阶贝塞尔波束在中心区域的电场渐渐集中, 强度增强, 与图3和图4所示的电场强度分布非常不同.

$图 5 斜入射时反向传播双零阶贝塞尔波束的电场强度分布　(a) ${C_1} = {C_2} = 5^\circ $; (b) ${C_1} = {C_2} = 15^\circ $; (c) ${C_1} = {C_2} = 30^\circ $\r\nFig. 5. Electric intensity distribution of back propagating double zero-order Bessel beams with oblique incidence: (a) ${C_1} = {C_2} = 5^\circ $; (b) ${C_1} = {C_2} = 15^\circ $; (c) ${C_1} = {C_2} = 30^\circ $.$

图 5 斜入射时反向传播双零阶贝塞尔波束的电场强度分布　(a)

${C_1} = {C_2} = 5^\circ$

; (b)

${C_1} = {C_2} = 15^\circ$

; (c)

${C_1} = {C_2} = 30^\circ$

Fig. 5. Electric intensity distribution of back propagating double zero-order Bessel beams with oblique incidence: (a)

${C_1} = {C_2} = 5^\circ$

; (b)

${C_1} = {C_2} = 15^\circ$

; (c)

${C_1} = {C_2} = 30^\circ$

下载: 全尺寸图片幻灯片

图6展示了不同半锥角下斜入射非反向传播的双零阶贝塞尔波束在 $xOy$ 面的电场强度分布情况. 计算参数为: ${\alpha _1} = 30^\circ$ , ${\alpha _2} = 120^\circ$ , ${\beta _1} = {\beta _2} = 0^\circ$ . 由于双零阶贝塞尔波束的传播方向不是反向, 电场强度在 $xOy$ 面的分布没有形成较为明显的孤岛, 这与非反向传播的波束并不会形成驻波的事实相符合. 随着半锥角的增大, 双零阶贝塞尔波束中心区域的电场渐渐分散, 与图5所示的电场强度分布刚好相反. 与图3—图5相比, 图6所展示的非反向传播的双零阶贝塞尔波束场强分布与反向传播的双零阶贝塞尔波束场强分布有着明显的区别, 这会导致入射粒子的散射特性的不同.

$图 6 斜入射时非反向传播双零阶贝塞尔波束的电场强度分布　(a) ${C_1} = {C_2} = 5^\circ $; (b) ${C_1} = {C_2} = 15^\circ $; (c) ${C_1} = $$ {C_2} = 30^\circ $\r\nFig. 6. Electric intensity distribution of non-back propagating double zero-order Bessel beams with oblique incidence: (a) ${C_1} = $$ {C_2} = 5^\circ $; (b) ${C_1} = {C_2} = 15^\circ $; (c) ${C_1} = {C_2} = 30^\circ $.$

图 6 斜入射时非反向传播双零阶贝塞尔波束的电场强度分布　(a)

${C_1} = {C_2} = 5^\circ$

; (b)

${C_1} = {C_2} = 15^\circ$

; (c)

${C_1} =$

${C_2} = 30^\circ$

Fig. 6. Electric intensity distribution of non-back propagating double zero-order Bessel beams with oblique incidence: (a)

${C_1} =$

${C_2} = 5^\circ$

; (b)

${C_1} = {C_2} = 15^\circ$

; (c)

${C_1} = {C_2} = 30^\circ$

下载: 全尺寸图片幻灯片

2.4 UA介质球的内场展开

对于半径为a的UA介质球, 其介电常数 ${\mathop {\boldsymbol{\varepsilon}} \limits^{=}}$ 和磁导率 ${\mathop {\boldsymbol{\mu}} \limits^{=}}$ 分别表示为

${\mathop {\boldsymbol{\varepsilon}} \limits^{=}} = \left[ {\begin{array}{*{20}{c}} {{\varepsilon _t}}&0&0 \\ 0&{{\varepsilon _t}}&0 \\ 0&0&{{\varepsilon _z}} \end{array}} \right] \text{, }\;\;{\mathop {\boldsymbol{\mu}} \limits^{=}} = \left[ {\begin{array}{*{20}{c}} {{\mu _t}}&0&0 \\ 0&{{\mu _t}}&0 \\ 0&0&{{\mu _z}} \end{array}} \right].$

(18)

在无源情况下, 由UA介质的麦克斯韦方程组可以得到电场矢量波动方程:

$\nabla \times ({{\mathop {\boldsymbol{\mu}} \limits^{=}}{}^{ - 1}} \cdot \nabla \times {\boldsymbol{E}}) - {\omega ^2}{\mathop {\boldsymbol{\varepsilon}} \limits^{=}} \cdot {\boldsymbol{E}} = 0 .$

(19)

(19)式即是无源UA介质中电场矢量所满足的微分波动方程. 该波动方程由于 ${\mathop {\boldsymbol{\varepsilon}} \limits^{=}}$ 和 ${\mathop {\boldsymbol{\mu}} \limits^{=}}$ 是张量, 电场的各个分量耦合在一起, 无法用分离变量法来求解, 可以引入傅里叶变换来求解^[23]:

${\boldsymbol{E}}(r) = \int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {{\boldsymbol{E}}(k){{\text{e}}^{{\text{i}}k \cdot r}}{\text{d}}{k_x}{\text{d}}{k_y}{\text{d}}{k_z}} } } .$

(20)

通过引入傅里叶变换解本征方程, UA介质球的内部电磁场可以用球矢量波函数表示为^[36]

$\begin{split}{{\boldsymbol{E}}}^{\text{I}}(r)=\;&\displaystyle \sum _{q=1}^{2}\displaystyle \sum _{n=1}^{\infty }\displaystyle \sum _{m=-n}^{n}\displaystyle \sum _{{n'}=1}^{\infty }2\text{π}{G}_{m{n'}q}\displaystyle \int _{0}^{\text{π}}\left[{A}_{mnq}^{\text{e}}{{\boldsymbol{M}}}_{mn}^{(1)}(r,{k}_{q}) +{B}_{mnq}^{\text{e}}{{\boldsymbol{N}}}_{mn}^{(1)}(r,{k}_{q})+{C}_{mnq}^{\text{e}}{{\boldsymbol{L}}}_{mn}^{(1)}(r,{k}_{q})\right] \\ & \times {{\rm{P}}}_{{n'}}^{m}(\mathrm{cos}{\theta }_{k}){k}_{q}^{2}\mathrm{sin}{\theta }_{k}\text{d}{\theta }_{k}\text{, }\end{split}$

(21)

$\begin{split}{{\boldsymbol{H}}}^{\text{I}}(r)=\;&\displaystyle \sum _{q=1}^{2}\displaystyle \sum _{n=1}^{\infty }\displaystyle \sum _{m=-n}^{n}\displaystyle \sum _{{n'}=1}^{\infty }2\text{π}{G}_{m{n'}q}\displaystyle \int_{0}^{\text{π}} \left[{A}_{mnq}^{\text{h}}{{\boldsymbol{M}}}_{mn}^{(1)}(r,{k}_{q}) +{B}_{mnq}^{\text{h}}{{\boldsymbol{N}}}_{mn}^{(1)}(r,{k}_{q})+{C}_{mnq}^{\text{h}}{{\boldsymbol{L}}}_{mn}^{(1)}(r,{k}_{q})\right]\\ & \times {{\rm{P}}}_{{n'}}^{m}(\mathrm{cos}{\theta }_{k}){k}_{q}^{2}\mathrm{sin}{\theta }_{k}\text{d}{\theta }_{k}\text{, }\end{split}$

(22)

其中 $A_{mnq}^{\text{e}}, B_{mnq}^{\text{e}}, C_{mnq}^{\text{e}}, A_{mnq}^{\text{h}}, B_{mnq}^{\text{h}}$ 和 $C_{mnq}^{\text{h}}$ 是介电常数张量展开系数, 其表达式详见文献[23], 注意表达式中的介电常数张量和磁导率等均与UA介质球保持一致. $G_{ {{mn}}'{{q} }}$ 是未知的内场展开系数, 由边界条件确定. 散射场在粒子坐标系下可以用球矢量波函数展开为

$\begin{split} {{{\boldsymbol{E}}}^{\text{s}}} = \sum\limits_{n = 1}^\infty {\sum\limits_{m = - n}^n {\left[ {A_{mn}^{\text{s}}{\boldsymbol{M}}_{mn}^{(3)}(r,{k_0}) + B_{mn}^{\text{s}}{\boldsymbol{N}}_{mn}^{(3)}(r,{k_0})} \right]} } \text{, } \end{split}$

(23)

$\begin{split} {{\boldsymbol{H}}^{\text{s}}} =\;& \frac{{{k_0}}}{{{\text{i}}\omega {\mu _0}}}\sum\limits_{n = 1}^\infty \sum\limits_{m = - n}^n {\left[ {A_{mn}^{\text{s}}{\boldsymbol{N}}_{mn}^{(3)}(r,{k_0}) + B_{mn}^{\text{s}}{\boldsymbol{M}}_{mn}^{(3)}(r,{k_0})} \right]} \text{, } \end{split}$

(24)

其中 $A_{mn}^{\text{s}}$ 和 $B_{mn}^{\text{s}}$ 为散射场展开系数.

2.5 通过边界条件求解散系数

在 $r = a$ 处, 由电磁场在边界上切向分量连续有

${ {\boldsymbol{E}}}\left| {_t = } \right.{{ {\boldsymbol{E}}}^{\text{i}}}\left| {_t} \right. + {{ {\boldsymbol{E}}}^{\text{s}}}\left| {_t} \right. \text{, }$

(25)

${\boldsymbol{H}}\left| {_t = } \right.{{\boldsymbol{H}}^{\text{i}}}\left| {_t} \right. + {{\boldsymbol{H}}^{\text{s}}}\left| {_t} \right. \text{, }$

(26)

其中t表示电场磁场的切向分量. 将入射场、内场及散射场的展开详细表达式代入边界条件, 于是在 $r = a$ 处, 可得

$\begin{split} &a_{mn}^{{\text{i}}t}{{\text{j}}_n}({k_0}r) + A_{mn}^{\text{s}}{\text{h}}_n^{(1)}({k_0}r) = \sum\limits_{q = 1}^2 \sum\limits_{n'} {2{\text{π}}{G_{mn'q}}}\int_0^{\text{π}} {A_{mnq}^{\text{e}}{{\text{j}}_n}({k_q}r)} {\text{P}}_{n'}^m(\cos {\theta _k})k_q^2\sin {\theta _k}{\text{d}}{\theta _k} \text{, } \end{split}$

(27)

$\begin{split}&{b}_{mn}^{\text{i}t}\frac{1}{{k}_{0}r}\frac{\text{d}\left[r{\text{j}}_{n}({k}_{0}r)\right]}{\text{d}r}+{B}_{mn}^{\text{s}}\frac{1}{{k}_{0}r}\frac{\text{d}\left[r{\text{h}}_{n}^{(1)}({k}_{0}r)\right]}{\text{d}r} \\ =\;&\displaystyle \sum _{q=1}^{2}{\displaystyle \sum _{{n'}}2\text{π}{G}_{m{n'}q}}\displaystyle {\int_0^{\text{π}}}\left\{{B}_{mnq}^{\text{e}}\frac{1}{{k}_{q}r}\frac{\text{d}\left[r{\text{j}}_{n}({k}_{q}r)\right]}{\text{d}r}+{C}_{mnq}^{\text{e}}\frac{{\text{j}}_{n}({k}_{q}r)}{r}\right\}{\text{P}}_{{n'}}^{m}(\mathrm{cos}{\theta }_{k}){k}_{q}^{2}\mathrm{sin}{\theta }_{k}\text{d}{\theta }_{k}\text{, }\end{split}$

(28)

$\begin{split}&\frac{{k}_{0}}{\text{i}\omega {\mu }_{0}}{a}_{mn}^{\text{i}t}\frac{1}{{k}_{0}r}\frac{\text{d}\left[r{\text{j}}_{n}({k}_{0}r)\right]}{\text{d}r}+\frac{{k}_{0}}{\text{i}\omega {\mu }_{0}}{A}_{mn}^{\text{s}}\frac{1}{{k}_{0}r}\frac{\text{d}\left[r{\text{h}}_{n}^{(1)}({k}_{0}r)\right]}{\text{d}r}\\ =\;& {\displaystyle \sum _{q=1}^{2}{\displaystyle \sum _{{n'}}2\text{π}{G}_{m{n'}q}}}\displaystyle {\int_0^{\text{π}}}\left\{{B}_{mnq}^{h}\frac{1}{{k}_{q}r}\frac{\text{d}\left[r{\text{j}}_{n}({k}_{q}r)\right]}{\text{d}r}+{C}_{mnq}^{h}\frac{{\text{j}}_{n}({k}_{q}r)}{r}\right\}{\text{P}}_{{n'}}^{m}(\mathrm{cos}{\theta }_{k}){k}_{q}^{2}\mathrm{sin}{\theta }_{k}\text{d}{\theta }_{k}\text{, }\end{split}$

(29)

$\begin{split}\frac{{k}_{0}}{\text{i}\omega {\mu }_{0}}{b}_{mn}^{\text{i}t}{\text{j}}_{n}({k}_{0}r)+\frac{{k}_{0}}{{\rm i}\omega {\mu }_{0}}{B}_{mn}^{s}{\text{h}}_{n}^{(1)}({k}_{0}r) ={\displaystyle \sum _{q=1}^{2}{\displaystyle \sum _{{n'}}2\text{π}{G}_{m{n'}q}{\displaystyle {\int_0^{\text{π}}}{A}_{mnq}^{h}{\text{j}}_{n}({k}_{q}r)}}} {\text{P}}_{{n'}}^{m}(\mathrm{cos}{\theta }_{k}){k}_{q}^{2}\mathrm{sin}{\theta }_{k}\text{d}{\theta }_{k}. \end{split}$

(30)

求解上述方程组(27)—(30), 消去散射系数可得

$\begin{split}{a}_{mn}^{\text{i}t}\frac{1}{{k}_{0}r}\left[{\text{j}}_{n}({k}_{0}r)\frac{\text{d}\left[r{\text{h}}_{n}^{(1)}({k}_{0}r)\right]}{\text{d}r}-{\text{h}}_{n}^{(1)}({k}_{0}r)\frac{\text{d}\left[r{\text{j}}_{n}({k}_{0}r)\right]}{\text{d}r}\right] ={\displaystyle \sum _{q=1}^{2}{\displaystyle \sum _{{n'}}2\text{π}{G}_{m{n'}q}{\displaystyle {\int_0^{\text{π}}}{Q}_{mnq}}}}{\text{P}}_{{n'}}^{m}(\mathrm{cos}{\theta }_{k}){k}_{q}^{2}\mathrm{sin}{\theta }_{k}\text{d}{\theta }_{k}\text{, }\end{split}$

(31)

$\begin{split}& {b}_{mn}^{\text{i}t}\frac{1}{{k}_{0}r} \left[{\text{j}}_{n}({k}_{0}r)\frac{\text{d} \left[r{\text{h}}_{n}^{(1)}({k}_{0}r) \right]}{\text{d}r} - {\text{h}}_{n}^{(1)}({k}_{0}r)\frac{\text{d} \left[r{\text{j}}_{n}({k}_{0}r )\right]}{\text{d}r} \right] ={\displaystyle \sum _{q=1}^{2}{\displaystyle \sum _{{n'}}2\text{π}{G}_{m{n'}q}{\displaystyle {\int_0^{\text{π}}}{R}_{mnq}}}}{\text{P}}_{{n'}}^{m}(\mathrm{cos}{\theta }_{k}){k}_{q}^{2}\mathrm{sin}{\theta }_{k}\text{d}{\theta }_{k}\text{, }\end{split}$

(32)

其中,

$\begin{split}{R}_{mnq}=\left\{\frac{\text{i}\omega {\mu }_{0}}{{k}_{0}}{A}_{mnq}^{\text{h}}\frac{1}{{k}_{0}r}\frac{\text{d}}{\text{d}r}\left[r{\text{h}}_{n}^{(1)}({k}_{0}r)\right]{\text{j}}_{n}({k}_{q}r)-\left[{B}_{mnq}^{\text{e}}\frac{1}{{k}_{q}r}\frac{\text{d}}{\text{d}r}\left[r{\text{j}}_{n}({k}_{q}r)\right]+{C}_{mnq}^{\text{e}}\frac{{\text{j}}_{n}({k}_{q}r)}{r}\right]{\text{h}}_{n}^{(1)}({k}_{0}r)\right\}_{r=a}\text{, }\end{split}$

(33)

$\begin{split}{Q}_{mnq}=\left\{{A}_{mnq}^{\text{e}}\frac{1}{{k}_{0}r}\frac{\text{d}}{\text{d}r}\left[r{\text{h}}_{n}^{(1)}({k}_{0}r)\right]{\text{j}}_{n}({k}_{q}r) -\frac{\text{i}\omega {\mu }_{0}}{{k}_{0}} \left[{B}_{mnq}^{\text{h}}\frac{1}{{k}_{q}r}\frac{\text{d}}{\text{d}r}[r{\text{j}}_{n}({k}_{q}r)]+{C}_{mnq}^{\text{h}}\frac{{\text{j}}_{n}({k}_{q}r)}{r}\right]{\text{h}}_{n}^{(1)}({k}_{0}r)\right\}_{r=a}. \end{split}$

(34)

化简后可以得到关于内场展开系数 $G_{ {{ {mr}}'{ {q}} }}$ 的方程组:

$\begin{split} \sum\limits_{q = 1}^2 \sum\limits_{n' = 1}^\infty 2{\text{π}}{G_{mn'q}}\int_0^{\text{π}} Q_{mnq}{{\text{P}}_{n'}^m}({ \cos}\theta_k)k_q^2\sin\theta_k{\rm d}\theta_k={a}_{mn}^{{\rm i}t}\dfrac{\rm i }{(k_0a)^2} , \end{split}$

(35)

$\begin{split} \sum\limits_{q = 1}^2 \sum\limits_{n' = 1}^\infty 2{\text{π}}{G_{mn'q}}\int_0^{\text{π}} R_{mnq}{{\text{P}}_{n'}^m}({ \cos}\theta_k)k_q^2\sin\theta_k{\rm d}\theta_k={b}_{mn}^{{\rm i}t}\dfrac{\rm i }{(k_0a)^2} . \end{split}$

(36)

联立方程组求解内场展开系数 $G_{ {{ {mr}}'{ {q}} }}$ 将其代入(27)式和(30)式, 可求解出散射系数 $A_{mn}^{\text{s}}$ , $B_{mn}^{\text{s}}$ 的表达式为^[37]

$\begin{split} A_{mn}^{\text{s}} =\frac{1}{{{\text{h}}_n^{(1)}({k_0}a)}}\Bigg[ \sum\limits_{n' = 1}^\infty {2{\text{π}}{G_{mn'q}}} \sum\limits_{q = 1}^2 \int_0^{\text{π}} {A_{mnq}^{\text{e}}{{\text{j}}_n}({k_q}a)}{\text{P}}_{n'}^m(\cos {\theta _k})k_q^2\sin {\theta _k}{\text{d}}{\theta _k} - a_{mn}^{{\text{i}}t}{{\text{j}}_n}({k_0}a) \Bigg] \text{, } \end{split}$

(37)

$\begin{split} B_{mn}^{\text{s}} = \frac{1}{{{\text{h}}_n^{(1)}({k_0}a)}}\Bigg[ \frac{{{\text{i}}\omega {\mu _0}}}{{{k_0}}}\sum\limits_{q = 1}^2 \sum\limits_{n' = 1}^\infty 2{\text{π}}{G_{mn'q}}\int_0^{\text{π}} A_{mnq}^h{{\text{j}}_n}({k_q}a){\text{P}}_{n'}^m(\cos {\theta _k}) k_q^2\sin {\theta _k}{\text{d}}{\theta _k} - b_{mn}^{{\text{i}}t}{{\text{j}}_n}({k_0}a) \Bigg] . \end{split}$

(38)

将散射系数代入(23)式和(24)式中得到散射场的电场强度. 由RCS的定义可得

$\begin{split}\sigma =\;&\underset{r\to \infty }{\mathrm{lim}}\left(4\text{π}{r}^{2} \dfrac{{\left|{{\boldsymbol{E}}}^{\text{s}}\right|}^2}{\left|{{\boldsymbol{E}}}^{\text{i}}\right|^{2}}\right)= \frac{4\text{π}}{{k}_{0}^{2}}\left\{\left|\displaystyle \sum _{n=1}^{\infty }\displaystyle \sum _{m=-n}^{n}{(-\text{i})}^{n}{\text{e}}^{\text{i}m\phi }\left[m{A}_{mn}^{\text{s}}\frac{{\text{P}}_{n}^{m}(\mathrm{cos}\theta )}{\mathrm{sin}\theta }+{B}_{mn}^{\text{s}}\frac{{\text{dP}}_{n}^{m}(\mathrm{cos}\theta )}{\text{d}\theta }\right]\right|^{2}\right.\\ &\left. +\left|\displaystyle \sum _{n=1}^{\infty }\displaystyle \sum _{m=-n}^{n}{(-\text{i})}^{n+1}{\text{e}}^{\text{i}m\phi } \left[{A}_{mn}^{\text{s}}\frac{{\text{dP}}_{n}^{m}(\mathrm{cos}\theta )}{\text{d}\theta }+m{A}_{mn}^{\text{s}}\frac{{\text{P}}_{n}^{m}(\mathrm{cos}\theta )}{\mathrm{sin}\theta }\right]\right|^{2}\right\}. \end{split}$

(39)

3. 数值结果和讨论

当半锥角为0时, 零阶贝塞尔波束将退化为平面波形式. 为了验证程序的正确性, 我们将单零阶贝塞尔波束退化成平面波, 计算了入射UA介质球RCS的角分布, 并且与文献[38]给出的结果进行了比较. 如图7所示, 发现二者结果符合较好, 验证了程序与理论公式推导的正确性. 图7使用的参数为: ${\varepsilon _t} = 2.5{\varepsilon _0}$ , ${\varepsilon _z} = 1.0{\varepsilon _0}$ , ${\mu _t} = {\mu _z} = 1.0{\mu _0}$ , $\lambda = 1.064\text{ }{{ \text{μm}}}$ , $\alpha = \beta = 0^\circ$ , $a = 1\text{ }{{ \text{μm}}}$ , ${C_0} = 0^\circ$ . E-plane对应xOz面, 称为E面; H-plane对应yOz面, 称为H面.

$图 7 UA介质球对单零阶贝塞尔波束的RCS角分布\r\nFig. 7. Normalized RCS of a UA dielectric sphere illuminated by single zero-order Bessel beam versus the scattering angle.$

图 7 UA介质球对单零阶贝塞尔波束的RCS角分布

Fig. 7. Normalized RCS of a UA dielectric sphere illuminated by single zero-order Bessel beam versus the scattering angle.

下载: 全尺寸图片幻灯片

图8计算了反向传输的双零阶贝塞尔波束退化成平面波入射到UA球形粒子的RCS的角分布, 并与文献[39]结果进行了对比, 发现两者也符合得比较好, 进一步验证了本文理论和程序的正确性. 计算过程中所使用的参数为 ${\varepsilon _t} = 2.5{\varepsilon _0}$ , ${\varepsilon _z} = 1.0{\varepsilon _0}$ , ${\mu _t} = {\mu _z} = 1.0{\mu _0}$ , $\lambda = 1.064\text{ }{{ \text{μm}}}$ , $a = 1\text{ }{{ \text{μm}}}$ , ${\alpha _1} = {\beta _1} = {\beta _2} = 0^\circ$ , ${\alpha _2} = 180^\circ$ , ${x_1} = {y_1} = {x_2} = {y_2} = 0$ , ${z_1} = 1\text{ }{{ \text{μm}}}$ , ${z_2} = - 1\text{ }{{ \text{μm}}}$ . 比较图7和图8可以看出, 当双零阶贝塞尔波束照射UA球形粒子时, 产生的散射效应是增加的. 双零阶贝塞尔波束入射情况下, 由于波束在粒子中存在干涉效应, 所以E面和H面的峰值也不是特别明显.

$图 8 双零阶贝塞尔波束退化成平面波入射UA介质球的RCS角分布　(a) E-plane; (b) H-plane\r\nFig. 8. Angular distribution of the RCS of a UA dielectric sphere by double zero-order Bessel beams when degenerating into plane waves: (a) E-plane; (b) H-plane.$

图 8 双零阶贝塞尔波束退化成平面波入射UA介质球的RCS角分布　(a) E-plane; (b) H-plane

Fig. 8. Angular distribution of the RCS of a UA dielectric sphere by double zero-order Bessel beams when degenerating into plane waves: (a) E-plane; (b) H-plane.

下载: 全尺寸图片幻灯片

如图9所示, 计算了UA介质球对不同离轴情形单零阶贝塞尔波束的RCS角分布, 计算参数为: $\lambda = 1.064\text{ }{{ \text{μm}}}$ , ${\varepsilon _t} = 5.3495$ , ${\varepsilon _z} = 4.9284$ , ${\mu _t} = {\mu _z} = 1$ , ${n_0} = 1.0$ , ${\alpha _1} = {\beta _1} = {0^{\circ}}$ , ${C_1} = {C_1} = {10^{\circ}}$ . 当入射波束沿x轴方向偏移时, E面和H面上的RCS值都在减小, E面出现了不对称的情况, 但是H面上的RCS的角分布对称性和形状都没有太大改变. 当单波束沿y轴方向偏移时, E面和H面上的RCS值也都在减小, 与沿x轴方向偏移时的情形不同, 此时E面上的RCS角分布的对称性和形状都没有太大的改变, 但是在H面出现不对称的情况. 之所以出现RCS值减小的情况, 是因为波束离轴入射粒子, 波束中心距离球心横向位置越远时, 波束照射球体的强度会越小. 而沿x轴和y轴偏移会出现不一样的情况, 是因为方位角为 $\phi = \arctan ({y \mathord{\left/ {\vphantom {y x}} \right. } x})$ , x和y的改变会影响到散射系数.

$图 9 UA介质球对不同离轴情形入射的单零阶贝塞尔波束的RCS角分布　(a) E-plane; (b) H-plane\r\nFig. 9. Angular distribution of the RCS of a UA dielectric sphere illuminated by different off-axis incident zero-order Bessel beams: (a) E-plane; (b) H-plane.$

图 9 UA介质球对不同离轴情形入射的单零阶贝塞尔波束的RCS角分布　(a) E-plane; (b) H-plane

Fig. 9. Angular distribution of the RCS of a UA dielectric sphere illuminated by different off-axis incident zero-order Bessel beams: (a) E-plane; (b) H-plane.

下载: 全尺寸图片幻灯片

如图10所示, 计算了UA介质球对不同离轴双零阶贝塞尔波束的RCS角分布, 计算参数为: $\lambda = 1.064\text{ }{{ \text{μm}}}$ , ${\varepsilon _t} = 5.3495$ , ${\varepsilon _z} = 4.9284$ , ${\mu _t} = {\mu _z} = 1$ , ${n_0} = 1.0$ , ${\alpha _1} = {\beta _1} = {\beta _2} = 0^\circ$ , ${\alpha _2} = 180^\circ$ , ${C_1} = {C_2} = 10^\circ$ . 可以明显地看到, 无论波束是沿x轴方向还是沿y轴方向偏离, RCS值都会减小. 这是因为离轴入射时波束中心偏离球心的横向位置越远, 波束照射到球体上的强度就会越小, 这与单波束的情况类似. 与图9所示单离轴零阶贝塞尔波束入射情况相比, 图10中RCS角分布发生了明显地改变, 尤其是双零阶贝塞尔波束沿正y轴和负y轴离轴入射UA介质球时, RCS出现了两个极小值点.

$图 10 UA介质球对离轴入射的双零阶贝塞尔波束的RCS角分布　(a) E-plane; (b) H-plane\r\nFig. 10. Angular distribution of the RCS of a UA dielectric sphere illuminated by different off-axis incident double zero-order Bessel beams: (a) E-plane; (b) H-plane.$

图 10 UA介质球对离轴入射的双零阶贝塞尔波束的RCS角分布　(a) E-plane; (b) H-plane

Fig. 10. Angular distribution of the RCS of a UA dielectric sphere illuminated by different off-axis incident double zero-order Bessel beams: (a) E-plane; (b) H-plane.

下载: 全尺寸图片幻灯片

图11计算了第二束零阶贝塞尔波束入射角与第一束零阶贝塞尔波束的不同时, 双波束非反向入射UA球形粒子的RCS角分布, 所用参数为: $\lambda = 1.064\text{ }{{ \text{μm}}}$ , ${\varepsilon _t} = 5.3495$ , ${\varepsilon _z} = 4.9284$ , ${\mu _t} = {\mu _z} = 1.0{\mu _0}$ , ${n_0} = 1.0$ , ${x_1} = {y_1} = {z_1} = 0$ , ${x_2} = {y_2} = {z_2} = 0$ , ${\alpha _1} = {\beta _1} = {\beta _2} = 0^\circ$ , ${C_1} = {C_2} = 20^\circ$ . 可以看出, 随着第二束零阶贝塞尔波束入射角不断的变化而产生的RCS角分布是不同的. 当入射角都是0°时, 两束零阶贝塞尔波束的传播方向相同, 此时RCS值最大. 但是两束波束由于在粒子之间有相互作用的关系, 所得的RCS值并不是单零阶贝塞尔波束入射UA球形粒子时RCS的2倍. 随着第二束零阶贝塞尔波束的入射角不断增大, 即第二束零阶贝塞尔波束是斜入射时, 在E面上峰值没有正入射时大, 并且峰值出现的次数会随着入射角的不断增大而增加. 这说明在双波束的入射角度变大时, 其相互干涉的效应不断增大. 而在H面上, 由于双零阶贝塞尔波束的极化方向垂直于H面, 所以H面上的RCS角分布总是对称的.

$图 11 UA介质球对不同入射角下双零阶贝塞尔波束的RCS角分布　(a) E-plane; (b) H-plane\r\nFig. 11. Angular distribution of the RCS of a UA dielectric sphere illuminated by double zero-order Bessel beams with different incident angles: (a) E-plane; (b) H-plane.$

图 11 UA介质球对不同入射角下双零阶贝塞尔波束的RCS角分布　(a) E-plane; (b) H-plane

Fig. 11. Angular distribution of the RCS of a UA dielectric sphere illuminated by double zero-order Bessel beams with different incident angles: (a) E-plane; (b) H-plane.

下载: 全尺寸图片幻灯片

图12计算了双反向传输的零阶贝塞尔波束在不同锥角入射的情况下UA介质球的RCS角分布, 参数如下: $\lambda = 1.064\text{ }{{ \text{μm}}}$ , $a = 0.5\text{ }{{ \text{μm}}}$ , ${\varepsilon _t} = 5.3495$ , ${\varepsilon _z} = 4.9284$ , ${\mu _t} = {\mu _z} = 1.0{\mu _0}$ , ${n_0} = 1.0$ , ${x_1} = {y_1} = {x_2} = {y_2} = 0$ , ${z_1} = {z_2} = 1\text{ }{{ \text{μm}}}$ , ${\alpha _1} = {\beta _1} = {\beta _2} = 0^\circ$ , ${\alpha _2} = 180^\circ$ . 可以看出, 随着零阶贝塞尔波束锥角的逐渐增大, 在E面和H面的RCS值都是逐渐减小. 这是因为随着锥角的增大, 波束之间的干涉效应在逐渐增加. 当两束波束都沿z轴传播时, 所得到的RCS值是对称的.

$图 12 双反向传输的零阶贝塞尔波束在不同锥角入射的情况下UA介质球的RCS角分布　(a) E-plane; (b) H-plane\r\nFig. 12. Angular distribution of the RCS of a UA dielectric sphere illuminated by back propagating double zero-order Bessel beams with different conic angles: (a) E-plane; (b) H-plane.$

图 12 双反向传输的零阶贝塞尔波束在不同锥角入射的情况下UA介质球的RCS角分布　(a) E-plane; (b) H-plane

Fig. 12. Angular distribution of the RCS of a UA dielectric sphere illuminated by back propagating double zero-order Bessel beams with different conic angles: (a) E-plane; (b) H-plane.

下载: 全尺寸图片幻灯片

4. 结　论

基于广义洛伦兹Mie理论, 研究了UA球形粒子对两束具有任意传播和极化方向的零阶贝塞尔波束的传播和散射特性. 利用坐标旋转关系及特殊函数的正交关系, 导出了任意传播和极化方向的双零阶贝塞尔波束的球矢量波函数的展开形式及展开系数的解析表达式, 详细讨论了单、双零阶贝塞尔波束的电场强度分布特性. 通过将单、双零阶贝塞尔波束退化成平面波的特殊情况下的RCS角分布与文献结果进行对比, 验证了本文理论及程序的正确性. 数值分析了各参数对双零阶贝塞尔波束入射UA球形粒子的RCS角分布的影响, 并与单零阶贝塞尔波束入射UA球形粒子的散射特性进行对比. 由于干涉效应, 当粒子被双零阶贝塞尔波束照射时, RCS值并不是单零阶贝塞尔波束入射时的两倍. 且第二束零阶贝塞尔波束在入射角增大的时候干涉效应逐渐增强, 说明使用双波束与各向异性介质产生的相互作用会更强. 本文理论和数值结果希望能为多波束入射下非各向同性、生物细胞等复杂粒子的散射特性、粒径分析以及光学微操控等提供理论依据.

References

[1]	Durnin J 1987 J. Opt. Soc. 4 651 Google Scholar
[2]	Garces-Chavez V, Roskey D, Summers M D, Melville H, Mcgloin D, Wright E M, Dholakia K 2004 Appl. Phys. Lett. 85 4001 Google Scholar
[3]	Marston P L 2006 J. Acoust. Soc. Am. 120 3518 Google Scholar
[4]	Zhang L, Marston P L 2012 J. Acoust. Soc. Am. 131 329 Google Scholar
[5]	Milne G, Dholakia K, Mcgloin D 2007 Opt. Express 15 13972 Google Scholar
[6]	Vahimaa P, Kettunen V, Kuittnen M 1997 J. Opt. Soc. Am. A 14 1817 Google Scholar
[7]	Ding D, Liu X 1999 J. Opt. Soc. Am. A 16 1286 Google Scholar
[8]	Cimár T, Kollárová V, Bouchal Z, Zemánek P 2006 New J. Phys. 8 43 Google Scholar
[9]	Taylor J M, Love G D 2009 J. Opt. Soc. Am. A 26 278 Google Scholar
[10]	Ambrosio L A, Herández-Figueroa H E 2011 Biomed. Opt. Express 2 1893 Google Scholar
[11]	Mishra S R 1991 Opt. Commun. 85 159 Google Scholar
[12]	Marton P L 2007 J. Acoust. Soc. Am. 121 753 Google Scholar
[13]	Ma X B, Li E 2010 Chin. Opt. Lett. 8 1195 Google Scholar
[14]	Gouesbet G, Maheu B, Gréhan G 1988 J. Opt. Soc. Am. A 5 1427 Google Scholar
[15]	Li R X, Guo L X, Ding C 2013 Opt. Commun. 307 25 Google Scholar
[16]	Mitri F G 2011 Opt. Lett. 36 766 Google Scholar
[17]	Cui Z W, Han Y P, Han L 2013 J. Opt. Soc. Am. A 30 1913 Google Scholar
[18]	Klimov V 2020 Opt. Lett. 45 4300 Google Scholar
[19]	Wang X, Wang L, Lin P 2021 J. Innovative. Opt. Health Sci. 14 2150008 Google Scholar
[20]	Stout B, Nevière M, Popov E 2006 J. Opt. Soc. Am. A 23 1111 Google Scholar
[21]	Wong K L, Chen H T 1992 IEE Proc. H 139 314 Google Scholar
[22]	Qiu C W, Li L W, Yeo T S 2007 Phys. Rev. E 75 026609 Google Scholar
[23]	Geng Y L, Wu X B, Li L W, Guan B R 2004 Phys. Rev. E 70 056609 Google Scholar
[24]	Wang M J, Zhang H Y, Liu G S, Li Y L 2012 J. Opt. Soc. Am. A 29 2376 Google Scholar
[25]	Yuan Q K, Wu Z S, Li Z J 2010 J. Opt. Soc. Am. A 27 1457 Google Scholar
[26]	Wu Z S, Yuan Q K, Peng Y, Li Z J 2009 J. Opt. Soc. Am. A 26 1778 Google Scholar
[27]	Wang J J, Chen A T, Han Y P 2015 J. Quant. Spectrosc. Radiat. Transfer. 167 135 Google Scholar
[28]	Qu T, Wu Z S, Shang Q C, Li Z J, Bai L 2013 J. Opt. Soc. Am. A 30 1661 Google Scholar
[29]	Zemánek P, Alexandr J, Liska M 2002 J. Opt. Soc. Am. A 19 1025 Google Scholar
[30]	Li Z J, Wu Z S, Qu T, Li H Y, Bai L, Gong L 2015 J. Quant. Spectrosc. Radiat. Transfer. 162 56 Google Scholar
[31]	Hulst H 1957 Phys. Today 10 28 Google Scholar
[32]	Barton J P, Alexander D R, Schaub S A 1988 J. Appl. Phys. 64 1632 Google Scholar
[33]	Li Z J, Wu Z S, Qu T, Shang Q C, Bai L 2016 J. Quant. Spectrosc. Radiat. Transfer. 169 1 Google Scholar
[34]	Bolton H C 1959 Mathematical Gazette 43 157 Google Scholar
[35]	Li Z J, Wu Z S, Huan L, Li H Y 2011 Chin. Phys. B 20 081101 Google Scholar
[36]	Li M M, Yin S H, Yao B L, Lei M, Yang Y L, Min J W 2015 J. Opt. Soc. Am. B 32 468 Google Scholar
[37]	Wu Z S, Li Z J, Li H Y, Yuan Q K, Li H Y 2011 IEEE Trans. Antennas Propag. 59 4740 Google Scholar
[38]	李正军, 吴振森, 屈檀, 白璐, 曹运华 2014 电波科学学报 29 668 Google Scholar Li Z J, Wu Z S, Qu T, Bai L, Cao Y H 2014 Chin. J. Radio 29 668 Google Scholar
[39]	Li Z J, Wu Z S, Shang Q C 2015 Procedia Eng. 102 89 Google Scholar

Cited By

期刊类型引用(0)

其他类型引用(1)

图 1 双零阶贝塞尔波束离轴斜入射UA介质球的示意图

Figure 1. Schematic diagram of a uniaxial anisotropic sphere illuminated by off-axis obliquely incident double zero-order Bessel beams.

DownLoad: Full-Size Img PowerPoint

图 2 两直角坐标的旋转关系示意图

Figure 2. Schematic diagram of rotation relationship between two rectangular coordinate systems.

DownLoad: Full-Size Img PowerPoint

图 3 相同极化角下反向传播双零阶贝塞尔波束的电场强度分布　(a) ${C_1} = {C_2} = 5^\circ$ ; (b) ${C_1} = {C_2} = 15^\circ$ ; (c) ${C_1} = {C_2} = 30^\circ$

Figure 3. Electric intensity distribution of back propagating double zero-order Bessel beams with identical polarization angles: (a) ${C_1} =$ ${C_2} = 5^\circ$ ; (b) ${C_1} = {C_2} = 15^\circ$ ; (c) ${C_1} = {C_2} = 30^\circ$ .

DownLoad: Full-Size Img PowerPoint

图 4 不同极化角下反向传播双零阶贝塞尔波束的电场强度分布　(a) ${C_1} = {C_2} = 5^\circ$ ; (b) ${C_1} = {C_2} = 15^\circ$ ; (c) ${C_1} = {C_2} = 30^\circ$

Figure 4. Electric intensity distribution of back propagating double zero-order Bessel beams with different polarization angles: (a) ${C_1} = {C_2} = 5^\circ$ ; (b) ${C_1} = {C_2} = 15^\circ$ ; (c) ${C_1} = {C_2} = 30^\circ$ .

DownLoad: Full-Size Img PowerPoint

图 5 斜入射时反向传播双零阶贝塞尔波束的电场强度分布　(a) ${C_1} = {C_2} = 5^\circ$ ; (b) ${C_1} = {C_2} = 15^\circ$ ; (c) ${C_1} = {C_2} = 30^\circ$

Figure 5. Electric intensity distribution of back propagating double zero-order Bessel beams with oblique incidence: (a) ${C_1} = {C_2} = 5^\circ$ ; (b) ${C_1} = {C_2} = 15^\circ$ ; (c) ${C_1} = {C_2} = 30^\circ$ .

DownLoad: Full-Size Img PowerPoint

图 6 斜入射时非反向传播双零阶贝塞尔波束的电场强度分布　(a) ${C_1} = {C_2} = 5^\circ$ ; (b) ${C_1} = {C_2} = 15^\circ$ ; (c) ${C_1} =$ ${C_2} = 30^\circ$

Figure 6. Electric intensity distribution of non-back propagating double zero-order Bessel beams with oblique incidence: (a) ${C_1} =$ ${C_2} = 5^\circ$ ; (b) ${C_1} = {C_2} = 15^\circ$ ; (c) ${C_1} = {C_2} = 30^\circ$ .

DownLoad: Full-Size Img PowerPoint

图 7 UA介质球对单零阶贝塞尔波束的RCS角分布

Figure 7. Normalized RCS of a UA dielectric sphere illuminated by single zero-order Bessel beam versus the scattering angle.

DownLoad: Full-Size Img PowerPoint

图 8 双零阶贝塞尔波束退化成平面波入射UA介质球的RCS角分布　(a) E-plane; (b) H-plane

Figure 8. Angular distribution of the RCS of a UA dielectric sphere by double zero-order Bessel beams when degenerating into plane waves: (a) E-plane; (b) H-plane.

DownLoad: Full-Size Img PowerPoint

图 9 UA介质球对不同离轴情形入射的单零阶贝塞尔波束的RCS角分布　(a) E-plane; (b) H-plane

Figure 9. Angular distribution of the RCS of a UA dielectric sphere illuminated by different off-axis incident zero-order Bessel beams: (a) E-plane; (b) H-plane.

DownLoad: Full-Size Img PowerPoint

图 10 UA介质球对离轴入射的双零阶贝塞尔波束的RCS角分布　(a) E-plane; (b) H-plane

Figure 10. Angular distribution of the RCS of a UA dielectric sphere illuminated by different off-axis incident double zero-order Bessel beams: (a) E-plane; (b) H-plane.

DownLoad: Full-Size Img PowerPoint

图 11 UA介质球对不同入射角下双零阶贝塞尔波束的RCS角分布　(a) E-plane; (b) H-plane

Figure 11. Angular distribution of the RCS of a UA dielectric sphere illuminated by double zero-order Bessel beams with different incident angles: (a) E-plane; (b) H-plane.

DownLoad: Full-Size Img PowerPoint

图 12 双反向传输的零阶贝塞尔波束在不同锥角入射的情况下UA介质球的RCS角分布　(a) E-plane; (b) H-plane

Figure 12. Angular distribution of the RCS of a UA dielectric sphere illuminated by back propagating double zero-order Bessel beams with different conic angles: (a) E-plane; (b) H-plane.

DownLoad: Full-Size Img PowerPoint

[1]	Durnin J 1987 J. Opt. Soc. 4 651 Google Scholar
[2]	Garces-Chavez V, Roskey D, Summers M D, Melville H, Mcgloin D, Wright E M, Dholakia K 2004 Appl. Phys. Lett. 85 4001 Google Scholar
[3]	Marston P L 2006 J. Acoust. Soc. Am. 120 3518 Google Scholar
[4]	Zhang L, Marston P L 2012 J. Acoust. Soc. Am. 131 329 Google Scholar
[5]	Milne G, Dholakia K, Mcgloin D 2007 Opt. Express 15 13972 Google Scholar
[6]	Vahimaa P, Kettunen V, Kuittnen M 1997 J. Opt. Soc. Am. A 14 1817 Google Scholar
[7]	Ding D, Liu X 1999 J. Opt. Soc. Am. A 16 1286 Google Scholar
[8]	Cimár T, Kollárová V, Bouchal Z, Zemánek P 2006 New J. Phys. 8 43 Google Scholar
[9]	Taylor J M, Love G D 2009 J. Opt. Soc. Am. A 26 278 Google Scholar
[10]	Ambrosio L A, Herández-Figueroa H E 2011 Biomed. Opt. Express 2 1893 Google Scholar
[11]	Mishra S R 1991 Opt. Commun. 85 159 Google Scholar
[12]	Marton P L 2007 J. Acoust. Soc. Am. 121 753 Google Scholar
[13]	Ma X B, Li E 2010 Chin. Opt. Lett. 8 1195 Google Scholar
[14]	Gouesbet G, Maheu B, Gréhan G 1988 J. Opt. Soc. Am. A 5 1427 Google Scholar
[15]	Li R X, Guo L X, Ding C 2013 Opt. Commun. 307 25 Google Scholar
[16]	Mitri F G 2011 Opt. Lett. 36 766 Google Scholar
[17]	Cui Z W, Han Y P, Han L 2013 J. Opt. Soc. Am. A 30 1913 Google Scholar
[18]	Klimov V 2020 Opt. Lett. 45 4300 Google Scholar
[19]	Wang X, Wang L, Lin P 2021 J. Innovative. Opt. Health Sci. 14 2150008 Google Scholar
[20]	Stout B, Nevière M, Popov E 2006 J. Opt. Soc. Am. A 23 1111 Google Scholar
[21]	Wong K L, Chen H T 1992 IEE Proc. H 139 314 Google Scholar
[22]	Qiu C W, Li L W, Yeo T S 2007 Phys. Rev. E 75 026609 Google Scholar
[23]	Geng Y L, Wu X B, Li L W, Guan B R 2004 Phys. Rev. E 70 056609 Google Scholar
[24]	Wang M J, Zhang H Y, Liu G S, Li Y L 2012 J. Opt. Soc. Am. A 29 2376 Google Scholar
[25]	Yuan Q K, Wu Z S, Li Z J 2010 J. Opt. Soc. Am. A 27 1457 Google Scholar
[26]	Wu Z S, Yuan Q K, Peng Y, Li Z J 2009 J. Opt. Soc. Am. A 26 1778 Google Scholar
[27]	Wang J J, Chen A T, Han Y P 2015 J. Quant. Spectrosc. Radiat. Transfer. 167 135 Google Scholar
[28]	Qu T, Wu Z S, Shang Q C, Li Z J, Bai L 2013 J. Opt. Soc. Am. A 30 1661 Google Scholar
[29]	Zemánek P, Alexandr J, Liska M 2002 J. Opt. Soc. Am. A 19 1025 Google Scholar
[30]	Li Z J, Wu Z S, Qu T, Li H Y, Bai L, Gong L 2015 J. Quant. Spectrosc. Radiat. Transfer. 162 56 Google Scholar
[31]	Hulst H 1957 Phys. Today 10 28 Google Scholar
[32]	Barton J P, Alexander D R, Schaub S A 1988 J. Appl. Phys. 64 1632 Google Scholar
[33]	Li Z J, Wu Z S, Qu T, Shang Q C, Bai L 2016 J. Quant. Spectrosc. Radiat. Transfer. 169 1 Google Scholar
[34]	Bolton H C 1959 Mathematical Gazette 43 157 Google Scholar
[35]	Li Z J, Wu Z S, Huan L, Li H Y 2011 Chin. Phys. B 20 081101 Google Scholar
[36]	Li M M, Yin S H, Yao B L, Lei M, Yang Y L, Min J W 2015 J. Opt. Soc. Am. B 32 468 Google Scholar
[37]	Wu Z S, Li Z J, Li H Y, Yuan Q K, Li H Y 2011 IEEE Trans. Antennas Propag. 59 4740 Google Scholar
[38]	李正军, 吴振森, 屈檀, 白璐, 曹运华 2014 电波科学学报 29 668 Google Scholar Li Z J, Wu Z S, Qu T, Bai L, Cao Y H 2014 Chin. J. Radio 29 668 Google Scholar
[39]	Li Z J, Wu Z S, Shang Q C 2015 Procedia Eng. 102 89 Google Scholar

[1]	Xiang Meng, He Piao, Wang Tian-Yu, Yuan Lin, Deng Kai, Liu Fei, Shao Xiao-Peng. Computational polarized colorful Fourier ptychography imaging: a novel information reuse technique of polarization of scattering light field. Acta Physica Sinica, 2024, 73(12): 124202. doi: 10.7498/aps.73.20240268
[2]	Chen Xin-Miao, Li Hai-Ying, Wu Tao, Meng Xiang-Shuai, Li Feng-Xia. Near-field electromagnetic scattering of Bessel vortex beam by metal target. Acta Physica Sinica, 2023, 72(10): 100302. doi: 10.7498/aps.72.20222192
[3]	Li Liang-Liang, Wang Xiao-Fang. Analytical model for scattering effect of energetic charged-particle beam in radiography of steep density gradient region. Acta Physica Sinica, 2022, 71(11): 115201. doi: 10.7498/aps.70.20212269
[4]	Li Liangliang, Wang Xiaofang. An analytical model for scattering effect in energetic charged-particle radiography of a steep density gradient region and the characteristics of the resulting modulation structures. Acta Physica Sinica, 2022, 0(0): 0-0. doi: 10.7498/aps.71.20212269
[5]	Zhang Na, Zhao Jian-Min, Chen Ke, Zhao Jun-Ming, Jiang Tian, Feng Yi-Jun. Independent dual-beam control based on programmable coding metasurface. Acta Physica Sinica, 2021, 70(17): 178102. doi: 10.7498/aps.70.20210344
[6]	Shi Ze-Lin, Cui Sheng-Cheng, Xu Qing-Shan. Modified analytic expression for the single-scattering phase function. Acta Physica Sinica, 2017, 66(18): 180201. doi: 10.7498/aps.66.180201
[7]	Fu Cheng-Hua. Analysis of optical scattering of micro-nano particles. Acta Physica Sinica, 2017, 66(9): 097301. doi: 10.7498/aps.66.097301
[8]	Ma Yan, Lin Shu-Yu, Xian Xiao-Jun. Volume pulsation and scattering of bubbles under the second Bjerknes force. Acta Physica Sinica, 2016, 65(1): 014301. doi: 10.7498/aps.65.014301
[9]	Zhuang Jia-Yan, Chen Qian, He Wei-Ji, Mao Tian-Yi. Imaging through dynamic scattering media with compressed sensing. Acta Physica Sinica, 2016, 65(4): 040501. doi: 10.7498/aps.65.040501
[10]	Bai Min, Xuan Rong-Xi, Song Jian-Jun, Zhang He-Ming, Hu Hui-Yong, Shu Bin. Hole scattering and mobility in compressively strained Ge/(001)Si1-xGex. Acta Physica Sinica, 2015, 64(3): 038501. doi: 10.7498/aps.64.038501
[11]	Yin Cheng, Xu Tian, Chen Bing-Yan, Han Qing-Bang. Polarization characteristics of the lattice resonance of metal nanoparticle array. Acta Physica Sinica, 2015, 64(16): 164202. doi: 10.7498/aps.64.164202
[12]	Zhang Hui-Yun, Liu Meng, Yin Yi-Heng, Wu Zhi-Xin, Shen Duan-Long, Zhang Yu-Ping. Study on scattering properties of the metal wire gating in a THz band based on Green function method. Acta Physica Sinica, 2013, 62(19): 194207. doi: 10.7498/aps.62.194207
[13]	Sun Wu, Deng Xiao-Jiu, Li Yao-Dong, Zhang Yong-Ming, Zheng Sai-Jing, Wang Wei-Miao. Mechanism of dual-wavelength anti-jamming photoelectric smoke-detection. Acta Physica Sinica, 2013, 62(3): 030201. doi: 10.7498/aps.62.030201
[14]	Wang Hai-Hua, Sun Xian-Ming. Multiple scattering of light by mixtures of two different aerosol types. Acta Physica Sinica, 2012, 61(15): 154204. doi: 10.7498/aps.61.154204
[15]	Zhao Tai-Fei, Ke Xi-Zheng. Monte Carlo simulations for non-line-of-sight ultraviolet scattering coverage area. Acta Physica Sinica, 2012, 61(11): 114208. doi: 10.7498/aps.61.114208
[16]	He Jing-Bo, Liu Zhong, Hu Sheng-Liang. Detection of weak signal based on the sea clutter scattering. Acta Physica Sinica, 2011, 60(11): 110208. doi: 10.7498/aps.60.110208
[17]	Liu Wen-Jun, Mao Hong-Yan, Fu Guo-Qing, Qu Shi-Liang. Temporal statistics of multiply scattered terahertz pulses in scattering medium. Acta Physica Sinica, 2010, 59(2): 913-917. doi: 10.7498/aps.59.913
[18]	Chen Xing, Xia Yun-Jie. The scattering of two-mode squeezed vacuum state and entangled coherent state through a one-dimensional potential barrier. Acta Physica Sinica, 2010, 59(1): 80-86. doi: 10.7498/aps.59.80
[19]	Wang Qing-Hua, Zhang Ying-Ying, Lai Jian-Cheng, Li Zhen-Hua, He An-Zhi. Application of Mie theory in biological tissue scattering characteristics analysis. Acta Physica Sinica, 2007, 56(2): 1203-1207. doi: 10.7498/aps.56.1203
[20]	Bai Lu, Wu Zhen-Sen, Chen Hui, Guo Li-Xin. Scattering of fundamental Gaussian beam from on-axis cluster spheres. Acta Physica Sinica, 2005, 54(5): 2025-2029. doi: 10.7498/aps.54.2025

期刊类型引用(0)
其他类型引用(1)

Catalog

Vol. 71, Issue 18 - 2022-09-20

Metrics

Abstract views: 5192
PDF Downloads: 65
Cited By: 1

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板