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近些年来，高斯波束在各向异性材料中传播特性一直受到持续性的关注，该研究不仅具有重要的理论价值，而且一些成果已广泛应用于雷达散射截面、光信号处理、微波器件、天线罩、光纤的优化设计和微带天线等领域。1972年，ALEXOPOULOS等人[1]研究了无限长非均匀介质圆柱对高斯波束片的散射。LOCK[2]研究了无限长均匀圆柱体对入射聚焦高斯光束的散射问题。GUO等人[3]提出一种有效而精确的递推算法来计算高斯波束垂直入射无限长多层圆柱的散射问题，并重点分析了圆柱的彩虹效应。随着广义洛伦兹-米理论的发展，HUANG和ZHANG等人[4-6]通过将圆柱散射场、内场和入射的高斯光束用适当的圆柱矢量波函数展开，应用电磁场边界条件，解析地解决了高斯光束在单轴各向异性圆柱体上的散射特性。CHEN和ZHANG等人[7]提出一种精确的半解析方法来计算回旋各向异性圆柱体对在轴高斯光束的散射问题。高斯波束的研究及其性质已经非常成熟[8-14]，只有少量文献中研究了其它高斯波束的性质[15-18]。
厄米-高斯波束是由高斯波束演化而来，具有重要的理论价值。作者在前人基础上，重点研究了厄米-高斯波束，首先讨论了电磁媒质的本构关系和分类，其次讨论了厄米-高斯波束在单轴各向异性圆柱中的散射特性，最后给出了数值结果和结论。
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任意电磁媒质的本构关系可表达为：
$ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{D}} = \overline{\overline \varepsilon } \cdot \mathit{\boldsymbol{E}} + \overline{\overline \xi } \cdot \mathit{\boldsymbol{H}}}\\ {\mathit{\boldsymbol{B}} = \overline{\overline \mu } \cdot \mathit{\boldsymbol{H}} + \overline{\overline \zeta } \cdot \mathit{\boldsymbol{E}}} \end{array}} \right. $
(1) 式中，D是电位移，E是电场强度，B是磁感应强度，H是磁场强度，$\boldsymbol{\overline{\overline \xi }} $和$\boldsymbol{\overline{\overline \zeta }}$为手征参量张量，$\boldsymbol{\overline{\overline \varepsilon }}$和$ \boldsymbol{\overline{\overline \mu }}$分别为媒质的介电常数张量和磁导率张量。当媒质的磁导率为标量，介电常数为张量时，称媒质为电各向异性媒质，反之为磁各向异性媒质。在直角坐标系Oxyz中，电各向异性媒质中的介电常数可表示为$\overline{\overline \varepsilon } =\hat{x}\hat{x}{{\varepsilon }_{1}}+\hat{y}\hat{y}{{\varepsilon }_{2}}+\hat{z}\hat{z}{{\varepsilon }_{3}}$，而D与E的关系可表示为：
$ \left[ {\begin{array}{*{20}{c}} {{D_x}}\\ {{D_y}}\\ {{D_z}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\varepsilon _1}}&0&0\\ 0&{{\varepsilon _2}}&0\\ 0&0&{{\varepsilon _3}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{E_x}}\\ {{E_y}}\\ {{E_z}} \end{array}} \right] $
(2) 式中，下标x, y, z表示电场的3个方向分量; $ \hat{x}, \hat{y}, \hat{z}$为电场的3个方向的单位矢量。若ε1，ε2，ε3全部相等，称为各向同性媒质; 若其中有两个相等，称为单轴各向异性媒质; 若3个对角元素均不相等，则称为双轴各向异性媒质。本文中主要研究单轴各向异性媒质ε1=ε2的情形。
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如图 1所示[4]，在直角坐标系Oxyz中有一个半径为r0的无限长单轴各向异性圆柱体，高斯波束在自由空间且沿直角坐标系Ox′y′z′的z′轴正方向传播，高斯波束的束腰半径为w0，束腰中心与圆心O′重合。波束的传播方向与z轴正方向夹角为β，在O′x′y′z′中点O坐标为(x0, y0, z0)。随时间变化的部分规定为exp(-iωt)。
Figure 1. The incidence of a Gaussian beam on an uniaxial anisotropic cylinder
正如参考文献[4]中所描述的，散射场用圆柱矢量波函数展开的表达式为：
$ {\mathit{\boldsymbol{E}}_{\rm{s}}} = {\mathit{\boldsymbol{E}}_0}\mathop \sum \limits_{m = - \infty }^\infty \mathop \smallint \nolimits_0^\pi [{\alpha _m}(\zeta ){\mathit{\boldsymbol{m}}_{m\lambda }}^{(3)} + {\beta _m}(\zeta ){\mathit{\boldsymbol{n}}_{m\lambda }}^{(3)}]{\mathit{\boldsymbol{e}}^{{\rm{i}}hz}}{\rm{d}}\zeta $
(3) $ \begin{array}{l} {\mathit{\boldsymbol{H}}_{\rm{s}}} = - {\rm{i}}{E_0}\frac{1}{{{\eta _0}}}\mathop \sum \limits_{m = - \infty }^\infty \mathop \smallint \nolimits_0^\pi [{\alpha _m}(\zeta ){\mathit{\boldsymbol{n}}_{m\lambda }}^{(3)} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\beta _m}(\zeta ){\mathit{\boldsymbol{m}}_{m\lambda }}^{(3)}]{{\rm{e}}^{{\rm{i}}hz}}{\rm{d}}\zeta \end{array} $
(4) 同样参考文献[4]中的单轴各向异性圆柱内部电磁场表示为：
$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{E}}_{\rm{w}}} = {E_0}\sum\limits_{q = 1}^2 {\sum\limits_{m = - \infty }^\infty {\int_0^\pi {{F_{mq}}} } } (\zeta )[{\alpha _{q, {\rm{e}}}}(\zeta ){\mathit{\boldsymbol{m}}_{m{\lambda _q}}}^{(1)} + }\\ {{\beta _{q, {\rm{e}}}}(\zeta ){\mathit{\boldsymbol{n}}_{m{\lambda _q}}}^{(1)} + {\gamma _{q, {\rm{e}}}}(\zeta ){\mathit{\boldsymbol{l}}_{m{\lambda _q}}}^{(1)}]{{\rm{e}}^{{\rm{i}}hz}}{\rm{d}}\zeta } \end{array} $
(5) $ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{H}}_{\rm{w}}} = - {\rm{i}}{E_0}\frac{1}{{{\eta _0}}}\sum\limits_{q = 1}^2 {\sum\limits_{m = - \infty }^\infty {\int_0^\pi {\frac{{{k_q}}}{{{k_0}}}} } } {F_{mq}}(\zeta ) \times \\ [{\beta _{q, {\rm{e}}}}(\zeta )(\zeta ){\mathit{\boldsymbol{m}}_{m{\lambda _q}}}^{(1)} + {\alpha _{q, {\rm{e}}}}(\zeta ){\mathit{\boldsymbol{n}}_{m{\lambda _q}}}^{(1)}]{{\rm{e}}^{{\rm{i}}hz}}{\rm{d}}\zeta \end{array} $
(6) (3) 式~(6)式中，Es, Hs表示散射场中电场和磁场，Ew, Hw表示圆柱内部的电场和磁场，λ=k0sinζ，h=k0cosζ，k0为自由空间中的波数，ζ为圆柱矢量波与坐标轴z方向的夹角; E0为电场振幅，η0为波阻抗; mmλq(j), nmλq(j), lmλq(j)是圆柱矢量波函数, j=1, 2, 3分别对应三类贝塞尔函数; 而αm(ζ)，βm(ζ)，Fm1(ζ)和Fm2(ζ)是待求的未知系数。可以令a12=ω2ε1μ0，a22=ω2ε3μ0，ε0和μ0是自由空间中的介电常数和磁导率，则式中其它参量为：k1=a1，k2=[a12a22+(a12-a22)k02cos2ζ]1/2/a1，λ1=a12－k02cos2ζ，λ2=a2a1-1a12－k02cos2ζ，α1, e(ζ)=1，β1, e(ζ)=γ1，e(ζ)=α2，e(ζ)=0，$ {{\beta }_{2, \text{e}}}(\zeta )=-\text{i}\times \frac{a_{1}^{2}{{a}_{2}}}{\sqrt{(a_{1}^{2}-k_{0}^{2}{{\cos }^{2}}\zeta )[a_{1}^{2}a_{2}^{2}+(a_{1}^{2}-a_{2}^{2})k_{0}^{2}{{\cos }^{2}}\zeta ]}}$, $ {{\gamma }_{2, \text{e}}}(\zeta )=-\frac{a_{1}^{2}-a_{2}^{2}}{a_{1}^{2}}\ \frac{{{a}_{1}}{{a}_{2}}{{k}_{0}}\cos \zeta \sqrt{a_{1}^{2}-k_{0}^{2}{{\cos }^{2}}\zeta }}{a_{1}^{2}a_{2}^{2}+(a_{1}^{2}-a_{2}^{2})k_{0}^{2}{{\cos }^{2}}\zeta }$
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电磁场的切向分量在r=r0的边界连续，则边界条件可以表示为：
$ \left\{ {\begin{array}{*{20}{l}} {\hat r \times ({\mathit{\boldsymbol{E}}_{\rm{s}}} + {\mathit{\boldsymbol{E}}_{\rm{i}}}) = \hat r \times {\mathit{\boldsymbol{E}}_{\rm{w}}}}\\ {\hat r \times ({\mathit{\boldsymbol{H}}_{\rm{s}}} + {\mathit{\boldsymbol{H}}_{\rm{i}}}) = \hat r \times {\mathit{\boldsymbol{H}}_{\rm{w}}}} \end{array}, (r = {r_0})} \right. $
(7) 式中，Ei和Hi表示入射电磁波束的电场和磁场。将(3)式~(6)式代入到(7)式中，边界条件可以写为：
$ \begin{array}{l} \hat r \times {E_0}\mathop \sum \limits_{m = - \infty }^\infty \mathop \smallint \nolimits_0^\pi [{\alpha _m}(\zeta ){\mathit{\boldsymbol{m}}_{m\lambda }}^{(3)} + {\beta _m}(\zeta ){\mathit{\boldsymbol{n}}_{m\lambda }}^{(3)}] \times \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\rm{e}}^{{\rm{i}}hz}}{\rm{d}}\zeta + \hat r \times {\mathit{\boldsymbol{E}}_{\rm{i}}}{|_{r = {r_0}}} = \\ \hat r \times {E_0}\sum\limits_{q = 1}^2 {\sum\limits_{m = - \infty }^\infty {\int_0^\pi {{F_{mq}}} } } (\zeta )[{\alpha _{q, {\rm{e}}}}(\zeta ){\mathit{\boldsymbol{m}}_{m{\lambda _q}}}^{(1)} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\beta _{q, {\rm{e}}}}(\zeta ){\mathit{\boldsymbol{n}}_{m{\lambda _q}}}^{(1)} + {\gamma _{q, {\rm{e}}}}(\zeta ){\mathit{\boldsymbol{l}}_{m{\lambda _q}}}^{(1)}]{\kern 1pt} {{\rm{e}}^{{\rm{i}}hz}}{\rm{d}}\zeta \end{array} $
(8) $ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat r \times {E_0}\mathop \sum \limits_{m = - \infty }^\infty \mathop \smallint \nolimits_0^\pi [{\alpha _m}(\zeta ){\mathit{\boldsymbol{n}}_{m\lambda }}^{(3)} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\beta _m}(\zeta ){\mathit{\boldsymbol{m}}_{m\lambda }}^{(3)}]{\kern 1pt} {{\rm{e}}^{{\rm{i}}hz}}{\rm{d}}\zeta + \hat r \times i{\eta _0}{\mathit{\boldsymbol{H}}_{\rm{i}}}{|_{r = {r_0}}} = \\ \hat r \times {E_0}\sum\limits_{q = 1}^2 {\sum\limits_{m = - \infty }^\infty {\int_0^\pi {\frac{{{k_q}}}{{{k_0}}}{F_{mq}}} } } (\zeta )[{\beta _{q, {\rm{e}}}}(\zeta )(\zeta ){\mathit{\boldsymbol{m}}_{m{\lambda _q}}}^{(1)} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\alpha _{q, {\rm{e}}}}(\zeta ){\mathit{\boldsymbol{n}}_{m{\lambda _q}}}^{(1)}]{\kern 1pt} {{\rm{e}}^{{\rm{i}}hz}}{\rm{d}}\zeta \end{array} $
(9) 遵循投影法的一般理论步骤，分别在(8)式和(9)式两边点乘$ \hat{z}{{\text{e}}^{-\text{i}{{h}_{1}}z}}{{\text{e}}^{-\text{i}{m}'\varphi }}$和$ \hat{\varphi }{{\text{e}}^{-\text{i}{{h}_{1}}z}}{{\text{e}}^{-\text{i}{m}'\varphi }}$，然后在圆柱面进行积分，可以得到未知的展开系数与Ei和Hi的关系式：
$ \begin{array}{l} \begin{array}{*{20}{c}} {\xi \frac{{\rm{d}}}{{{\rm{d}}\xi }}{{\rm{H}}_m}^{(1)}(\xi ){\alpha _m}(\zeta ) + \frac{{hm}}{{{k_0}}}{{\rm{H}}_m}^{(1)}(\xi ){\beta _m}(\zeta ) - }\\ {{F_{m1}}(\zeta ){\xi _1}\frac{{\rm{d}}}{{{\rm{d}}{\xi _1}}}{{\rm{J}}_m}({\xi _1}) - {F_{m2}}(\zeta ) \times } \end{array}\\ \begin{array}{*{20}{l}} {\left[ {{\beta _{2, {\rm{e}}}}(\zeta )\frac{{hm}}{{{k_2}}}{{\rm{J}}_m}({\xi _2}) - {\gamma _{2, {\rm{e}}}}(\zeta ){\rm{i}}m{{\rm{J}}_m}({\xi _2})} \right] = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\left( {\frac{1}{{2\pi }}} \right)}^2}\frac{1}{{{\mathit{\boldsymbol{E}}_0}}}\xi \int_{ - \infty }^\infty {\rm{d}} z\int_0^{2\pi } {\hat r} \times {\mathit{\boldsymbol{E}}_i} \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat z{\rm{exp}}( - {\rm{i}}m\varphi ){\rm{exp}}( - {\rm{i}}hz){\rm{d}}\varphi } \end{array} \end{array} $
(10) $ \begin{array}{l} \begin{array}{*{20}{c}} {{\xi ^2}{\mathit{\boldsymbol{H}}_m}^{(1)}(\xi ){\beta _m}(\zeta ) - {F_{m2}}(\zeta )\frac{{{k_0}}}{{{k_2}}}{\xi _2}^2{{\rm{J}}_m}({\xi _2}) \times }\\ {\left[ {{\beta _{2, {\rm{e}}}}(\zeta ) + {\gamma _{2, {\rm{e}}}}(\zeta )\frac{{{\rm{i}}h{k_2}}}{{\lambda _2^2}}} \right] = {{\left( {\frac{1}{{2\pi }}} \right)}^2}\frac{1}{{{E_0}}}{{({k_0}{r_0})}^2} \times } \end{array}\\ \begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{sin}}\zeta \int_{ - \infty }^\infty {\rm{d}} z\int_0^{2\pi } {\hat r} \times {\mathit{\boldsymbol{E}}_{\rm{i}}} \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat \varphi {\rm{exp}}( - {\rm{i}}m\varphi ){\rm{exp}}( - {\rm{i}}hz){\rm{d}}\varphi } \end{array} \end{array} $
(11) $ \begin{array}{*{20}{c}} {\frac{{hm}}{{{k_0}}}{\mathit{\boldsymbol{H}}_m}^{(1)}(\xi ){\alpha _m}(\zeta ) + \xi \frac{{\rm{d}}}{{{\rm{d}}\xi }}{\mathit{\boldsymbol{H}}_m}^{(1)}(\xi ){\beta _m}(\zeta ) - }\\ {\frac{{hm}}{{{k_0}}}{F_{m1}}(\zeta ){{\rm{J}}_m}({\xi _1}) - }\\ {\frac{{{k_2}}}{{{k_0}}}{F_{m2}}(\zeta ){\beta _{2, {\rm{e}}}}(\zeta ){\xi _2}\frac{{\rm{d}}}{{{\rm{d}}{\xi _2}}}{{\rm{J}}_m}({\xi _2}) = }\\ {{\rm{i}}{\eta _0}\frac{1}{{{E_0}}}{{\left( {\frac{1}{{2\pi }}} \right)}^2}\xi \int_{ - \infty }^\infty {\rm{d}} z\int_0^{2\pi } {\hat r} \times {\mathit{\boldsymbol{H}}_{\rm{i}}} \cdot \hat z{{\rm{e}}^{ - {\rm{i}}m\varphi }}{{\rm{e}}^{ - {\rm{i}}hz}}{\rm{d}}\varphi )} \end{array} $
(12) $ \begin{array}{*{20}{c}} {{\xi ^2}{\mathit{\boldsymbol{H}}_m}^{(1)}(\xi ){\alpha _m}(\zeta ) - {F_{m1}}(\zeta ){\xi _1}^2{{\rm{J}}_m}({\xi _1}) = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{i}}{\eta _0}\frac{1}{{{E_0}}}{{\left( {\frac{1}{{2\pi }}} \right)}^2}{{({k_0}{r_0})}^2}{\rm{sin}}\zeta \int_{ - \infty }^\infty {\rm{d}} z\int_0^{2\pi } {\hat r} \times }\\ {{\mathit{\boldsymbol{H}}_{\rm{i}}} \cdot \hat \varphi {\rm{exp}}( - {\rm{i}}m\varphi ){\rm{exp}}( - {\rm{i}}hz){\rm{d}}\varphi } \end{array} $
(13) 式中，ξ=λr0，ξ1=λ1r0和ξ2=λ2r0。
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DAVIS和BARTON给出了一种计算方法[19-20]，高斯波束TEM00(y′)或TE模在坐标系O′x′y′z′中电磁分量中各阶近似描述可表示为：
$ {{E_{{x^\prime }}} = {E_0}{s^2}( - 2{Q^2}\xi \eta ){\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}}} $
(14) $ \begin{array}{l} {E_{{y^\prime }}} = {E_0}[1 + {s^2}({\rm{i}}{Q^3}{\rho ^4} - {Q^2}{\rho ^2} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2{Q^2}{\eta ^2})]{\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}} \end{array} $
(15) $ \begin{array}{*{20}{c}} {{E_{{z^\prime }}} = {E_0}[2sQ\eta + {s^3}(2{\rm{i}}{Q^4}{\rho ^4}\eta - }\\ {6{Q^3}{\rho ^2}\eta )]{\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}}} \end{array} $
(16) $ \begin{array}{*{20}{c}} {{H_{{x^\prime }}} = - \frac{{{E_0}}}{\eta }[1 + {s^2}({\rm{i}}{Q^3}{\rho ^4} - }\\ {{Q^2}{\rho ^2} - 2{Q^2}{\xi ^2})]{\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}}} \end{array} $
(17) $ {H_{{y^\prime }}} = \frac{{{E_0}}}{\eta }{s^2}2{Q^2}\xi \eta {\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}} $
(18) $ \begin{array}{*{20}{l}} {{H_{{z^\prime }}} = - \frac{{{E_0}}}{\eta }[2sQ\xi + {s^3}( - 6{Q^3}{\rho ^2}\xi + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2{\rm{i}}{Q^4}{\rho ^4}\xi )]{\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}}} \end{array} $
(19) 式中，s=1/(kw0)，ψ0(ξ, η, ζ)=iQexp(－iρ2Q)，Q=1/(i－2ζ)，ρ2=ξ2+η2。
根据电磁场理论中的对偶关系，即在(14)式~(19)式中做如下替换:E→－H，H→E，ε0→μ0，μ0→ε0，得到的电磁场分量仍然近似满足麦克斯韦方程组，是高斯波束的TEM00(x′)或TM模。
正如参考文献[19]中提到的，厄米-高斯波束的各种模式可以通过求解TEM00(y′)或TEM00(x′)的偏导数得到，即：
$ {\rm{TE}}{{\rm{M}}_{mn}}({y^\prime }) = \frac{{{\partial ^m}{\partial ^n}{\rm{TE}}{{\rm{M}}_{00}}^{({y^\prime })}}}{{\partial {\xi ^m}\partial {\eta ^n}}} $
(20) $ {\rm{TE}}{{\rm{M}}_{mn}}({x^\prime }) = \frac{{{\partial ^m}{\partial ^n}{\rm{TE}}{{\rm{M}}_{00}}^{({x^\prime })}}}{{\partial {\xi ^m}\partial {\eta ^n}}} $
(21) 式中，ξ和η为无量纲的参量：ξ=x′/w0，η=y′/w0，w0为高斯波束束腰半径。
为了得到厄米-高斯波束的具体表达式，以TEM01(y′)为例，即在(20)式中令m=0, n=1可得:
$ {\rm{TE}}{{\rm{M}}_{01}}({y^\prime }) = \frac{{\partial {\rm{TE}}{{\rm{M}}_{00}}^{({y^\prime })}}}{{\partial \eta }} $
(22) $ {E_{{x^\prime }}} = {E_0}{s^2}( - 2{Q^2}\xi + 4{\rm{i}}{Q^3}\xi {\eta ^2}){\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}} $
(23) 将(14)式~(19)式代入(22)式，即可求得TEM01(y′)模：
$ \begin{array}{*{20}{c}} {{E_{{y^\prime }}} = {E_0}\{ - 2{\rm{i}}Q + {s^2}[2{\rm{i}}{Q^3}(5{\eta ^2} + 3{\xi ^2}) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 6{Q^2} + 2{Q^4}{\rho ^4}]\eta {\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}}} \end{array} $
(24) $ \begin{array}{*{20}{l}} {{E_{{z^\prime }}} = {E_0}\{ s(2Q - 4{\rm{i}}{Q^2}{\eta ^2}) + {s^3}[ - {Q^3}(18{\eta ^2} + 6{\xi ^2}) + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{i}}{Q^4}{\rho ^2}(22{\eta ^2} + 2{\xi ^2}) + 4{Q^5}{\rho ^4}{\eta ^2})\} {\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}}} \end{array} $
(25) $ \begin{array}{l} {H_{{x^\prime }}} = \frac{{{E_0}}}{\eta }\{ 2{\rm{i}}Q + {s^2}[2{Q^2} - {\rm{i}}{Q^3}(10{\xi ^2} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 6{\eta ^2}) - 2{Q^4}{\rho ^4}]\} \eta {\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}} \end{array} $
(26) $ {H_{{y^\prime }}} = \frac{{{E_0}}}{\eta }{s^2}2{Q^2}\xi (1 - 2{\rm{i}}Q{\eta ^2}){\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}} $
(27) $ \begin{array}{*{20}{l}} {{H_{{z^\prime }}} = - \frac{{{E_0}}}{\eta }\{ s( - 4{\rm{i}}{Q^2}) + {s^3}[ - 12{Q^3} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 20{\rm{i}}{Q^4}{\rho ^2} + 4{Q^5}{\rho ^4}]\} \xi \eta {\psi _0}{{\rm{e}}^{{\rm{i}}\zeta /{s^2}}}} \end{array} $
(28) (23) 式~(28)式即为厄米-高斯波束的具体表达式。类似可得出厄米-高斯波束的TEM10(x′)模。
图 2代表的是厄米-高斯波束TEM01(y′)的强度分布图, 图 3代表的是厄米-高斯波束TEM10(x′)的强度分布图。图中横纵坐标ξ，η均是无量纲的参量。
Figure 2. The intensity distribution of the TEM10(y′) mode Hermite-Gaussian beam
Figure 3. The intensity distribution of the TEM10(x′) mode Hermite-Gaussian beam
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入射的波束，其Ei和Hi可由(23)式~(28)式得到，代入到(10)式~(13)式，可以得到未知展开系数αm(ζ)，βm(ζ), Fm1(ζ)和Fm2(ζ)构成的线性方程组，从而求出这些未知系数。求出这些系数后再代入到(3)式~(6)式，进而可以求出散射场和内场。
定义归一化内场和近场的强度分布如下：
$ |{\mathit{\boldsymbol{E}}_{\rm{w}}}/{\mathit{\boldsymbol{E}}_0}{|^2} = |{\mathit{\boldsymbol{E}}_{{\rm{w}}, r}}{|^2} + |{E_{{\rm{w}}, \varphi }}{|^2} + |{E_{{\rm{w}}, z}}{|^2} $
(29) $ \begin{array}{*{20}{c}} {|({\mathit{\boldsymbol{E}}_{\rm{i}}} + {\mathit{\boldsymbol{E}}_{\rm{s}}})/{E_0}{|^2} = |{E_{{\rm{i}}, r}} + {E_{{\rm{s}}, r}}{|^2} + }\\ {|{E_{{\rm{i}}, \varphi }} + {E_{{\rm{s}}, \varphi }}{|^2} + |{E_{{\rm{i}}, z}} + {E_{{\rm{s}}, z}}{|^2}} \end{array} $
(30) 式中，Ew, r, Ew, φ, Ew, z分别为圆柱内部场中电场的3个分量; Ei, r, Ei, φ, Ei, z分别为入射场中电场的3个分量; Es, r, Es, φ, Es, z分别为散射场中电场的3个分量。
对于高斯波束入射的情形，使用的参量与模型是：单轴各向异性圆柱a1=3k0，a2=2k0, 高斯波束的束腰半径w0为2倍入射高斯波束的波长，圆柱的半径为5倍入射高斯波束的波长，入射角β=π/4，z0=0。图 4表示高斯波束TE模通过单轴各向异性圆柱的归一化内场和近场。其中对于x(wavelength)轴上的范围，5~15表示入射场，-5~5表示圆柱的内场，-15~-5表示透射场，图中色柱表示的物理量是无量纲的。
Figure 4. The normalized internal-field and near-field of a TE mode Gaussian beam through an uniaxial anisotropic cylinder
通过与已有方法及结果[4]比较可以发现，两者实现了很好的吻合，这在很大程度上验证了作者方法的正确性。
使用相同的参量和模型，可得厄米-高斯波束入射的情形，图 5和图 6分别表示厄米高斯波束TEM10(x′)模和TEM01(y′)通过单轴各向异性圆柱的归一化内场和近场。
Figure 5. The normalized internal-field and near-field of a TEM10(x′) mode Hermite-Gaussian beam through an uniaxial anisotropic cylinder
Figure 6. The normalized internal-field and near-field of a TEM10(y′) mode Hermite-Gaussian beam through an uniaxial anisotropic cylinder
从图 5可以看出，TEM10(x′)模式厄米-高斯波束入射单轴各向异性圆柱时的反射场强度很弱。圆柱类似于凸透镜，有一个会聚作用，所以波束在通过圆柱后的近场强度明显增强，由入射波和反射波叠加而成的驻波现象也同样在图中表现的非常明显。在图 6中，波束透过圆柱后内部场强度逐渐增强，同样也有驻波现象。比较两图形，发现在相同的情况下TEM10(x′)模式厄米-高斯波束通过单轴各向异性圆柱时的近场强度比TEM01(y′)模式的强，而反射场强度弱。另一个值得注意的现象是，TEM10(x′)模式的厄米-高斯波束在通过单轴各向异性圆柱时有一个明显的折射现象，而在TEM01(y′)模式厄米-高斯波束入射时则表现的不明显。
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主要基于单轴各向异性圆柱对厄米-高斯波束的散射特性进行研究。应用电磁场边界条件和投影法，精确半解析地得到了单轴各向异性圆柱对厄米-高斯波束散射特性和内场以及近场的归一化强度分布图，分析对比了两种不同的厄米高斯波束入射情形，发现波束透过圆柱后都有驻波现象。在相同的情况下TEM10(x′)模式厄米-高斯波束通过单轴各向异性圆柱时的近场强度比TEM01(y′)模式的强，而前者在通过单轴各向异性圆柱时折射现象更明显。
厄米-高斯波束对各向异性圆柱的散射特性研究
Scattering characteristics of Hermite-Gaussian beam on anisotropic cylinder
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摘要: 为了研究厄米-高斯波束在各向异性媒质中的散射特性，采用了将各向异性圆柱的散射场和内部场用圆柱矢量波函数展开，应用电磁场边界条件和投影法，提出了一种分析单轴各向异性圆柱对厄米-高斯波束散射特性研究的精确半解析方法; 获得了厄米-高斯波束通过单轴各向异性圆柱的内场以及近场的归一化强度分布图; 对两种不同的厄米高斯波束入射情形做出了分析和对比。结果表明，两种波束在通过圆柱后都有入射波和反射波叠加而成的驻波现象，而TEM10(x′)模式厄米-高斯波束入射后近场强度增强，且有明显的折射现象。该研究结果对厄米-高斯波束的应用具有一定的参考价值。Abstract: In order to research the scattering properties of Hermite-Gaussian beams in anisotropic media, the paper used a cylindrical vector wave function for the scattering field and internal field of an anisotropic cylinder. Using electromagnetic field boundary conditions and projection method, a method was proposed. The accurate semi-analytical method for studying the scattering properties of Hermite-Gaussian beams from uniaxial anisotropic cylinders was analyzed. The normalized intensity distributions of both the internal-field and near fields of the Hermitian-Gaussian beams through an uniaxial anisotropic cylinder were obtained. The analysis and comparison of two different Hermitage beam incidents were carried out. The results show that both beams have a standing wave phenomenon caused by the superposition of incident and reflected waves after passing through the cylinder, while the TEM10(x′) mode Hermite-Gaussian beam has a enhanced near-field intensity and a significant refraction phenomenon after its incidence. The research results have certain reference value for the application of Hermite-Gaussian beam.
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Figure 4. The normalized internal-field and near-field of a TE mode Gaussian beam through an uniaxial anisotropic cylinder
Figure 5. The normalized internal-field and near-field of a TEM10(x′) mode Hermite-Gaussian beam through an uniaxial anisotropic cylinder
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