Whispering gallery mode in a two-dimensional electromagnetic Helmholtz cavity
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摘要: 基于严格的广义双级数方法, 研究了二维电磁亥姆霍兹腔内回音壁模式的激发; 在给出几种回音壁模式的激发波长的同时, 探讨了入射角、亥姆霍兹腔开口大小对回音壁模式的影响. 结果表明, 回音壁模式对入射波长和腔开口大小的改变非常敏感, 但相比而言, 固定腔的方位角不变时, 入射角在较宽范围内的变化并不影响模式的激发. 因此, 腔的开口大小对亥姆霍兹腔的性能有很大影响, 是亥姆霍兹腔设计中需要重点关注的参量. 而当这些亥姆霍兹腔用于构建人工电磁材料时, 由于回音壁模式可在较宽的入射角下激发, 因而不需要严格要求每个腔的开口方向, 材料的加工难度也不大.Abstract: In this paper, whispering gallery mode (WGM) excited in a two-dimensional electromagnetic Helmholtz cavity are studied using a rigorous, generalized dual series approach. The excitation wavelengths of several whispering gallery modes are given, and the dependence of electromagnetic whispering gallery modes on the angle of incidence and the angular width of opening cavities is investigated. It was found that WGM are very sensitive to slight changes in wavelength or the angular width of the opening; at the same time, WGM can be excited across a wide range of incident angles given a fixed orientation angle of the cavity. This shows that the angular width of the opening has a significant influence on the performance of Helmholtz cavities and hence is a key parameter in their design. On the other hand, given the lack of sensitivity to the incident angle, no particular specification is needed when designing an artificially structured electromagnetic material using these Helmholtz cavities; accordingly, the fabrication difficulty is relatively low.
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图 1 在一个封闭的腔内激发的回音壁模式的示意图
Fig. 1 Schematic demonstration of whispering gallery modes excited in a closed cavity
图 2 半径为r, 腔体的开口大小为θ的亥姆霍兹腔的几何结构图, 入射角为
$ {\theta }_{i} $ , 腔的开口中心线与x轴重合Fig. 2 Geometry of an EM Helmholtz cavity with radius r and angular width of the opening θ, θi is the angle of incidence and the center line of the opening is the x axis
图 3 半径r = 60 mm, 开口大小为
$\theta = 14^\circ $ 的亥姆霍兹腔, 在入射角${\theta _i} = {0^\circ }$ 时,前5个Mie散射系数的绝对量$|{B_m}|$ 与入射波长的关系Fig. 3 The absolute value of the first five Mie scattering coefficients,
$|{B_m}|$ , as a function of wavelength for a Helmholtz cavity with r = 60 mm and$\theta = 14^\circ $ , the angle of incidence is${\theta _i} = {0^\circ }$ 
图 4 半径
$r = 60$ mm, 开口角宽度不同的亥姆霍兹腔, 在入射角${\theta _i} = {0^\circ}$ 时, a) Mie散射系数的绝对值$|{B_1}|$ 和b)$|{B_3}|$ 随波长的变化; c) 偶极回音壁模式和d) 六极回音壁模式的Q值随腔的开口角宽度的变化Fig. 4 The absolute value of the Mie scattering coefficients a) and b) versus incident wavelength for Helmholtz cavities with the same radius r = 60 mm but different angular widths of the opening; The quality factor, Q, of the dipole WGM c) and the hexapole WGM d) versus incident wavelength for Helmholtz cavities with the same radius r = 60 mm but different angular widths of the opening. The angle of incidence is
${\theta _i} = {0^\circ}$ 
图 5 半径
$r = 60$ mm, 开口角宽度$\theta = {20^\circ}$ 的亥姆霍兹腔, 在波长a)$\lambda = 53.8\;{\rm{mm}}$ 和b)$\lambda = 59.2\;{\rm{mm}}$ 的波的入射下的电场分布图; c)、e)分别为半径$r = 60$ mm, 开口角宽度$\theta = {8^\circ}$ 的亥姆霍兹腔, 在入射波长$\lambda = 53.8\;{\rm{mm}}$ 的波的入射下的电场分布图和磁场矢量分布图; d)和f )分别为半径$r = 60$ mm, 开口角宽度$\theta = {10^\circ}$ 的亥姆霍兹腔, 在入射波长$\lambda = 59.2{\rm{mm}}$ 的波的入射下的电场分布图和磁场矢量分布图. 入射角均为${\theta _i} = 0^\circ $ Fig. 5 The E field distribution when an EM wave with a)
$\lambda = 53.8\;{\rm{mm}}$ and b)$\lambda = 59.2\;{\rm{mm}}$ strikes a Helmholtz cavity with radius r = 60 mm and angle width of the opening$\theta = {20^\circ}$ ; c) The E field distribution and e) the magnetic field vector distribution when an EM wave with$\lambda = 53.8\;{\rm{mm}}$ strikes a Helmholtz cavity with radius r = 60 mm and angle width of the opening$\theta = {8^\circ}$ ; d) The E field distribution f) and the magnetic field vector distribution when an EM wave with$\lambda = 59.2\;{\rm{mm}}$ strikes a Helmholtz cavity with radius r = 60 mm and angle width of the opening$\theta = 10^\circ $ . The angle of incidence is${\theta _i} = {0^\circ}$ 
图 6 a)半径
$r = 60$ mm, 开口角宽度$\theta = {8^\circ}$ 的亥姆霍兹腔, 在不同入射角下, Mie散射系数的绝对量$|{B_1}|$ 随波长的变化关系; b)半径$r = 60$ mm, 开口角宽度$\theta = {10^\circ}$ 的亥姆霍兹腔, 在不同入射角下, Mie散射系数的绝对量$|{B_3}|$ 随波长的变化关系; 半径$r = 60$ mm, 开口角宽度$\theta = {10^\circ}$ 的亥姆霍兹腔, 在入射波长$\lambda = 59.2\;{\rm{mm}}$ 、入射角分别为c) 15°和d) 25°时, 电场的分布图Fig. 6 a) The absolute value of the Mie scattering coefficients,
$|{B_1}|$ , versus incident wavelength for Helmholtz cavities with radius r = 60 mm and angular width of the opening$\theta = {8^\circ}$ at different incident angle; b) The absolute value of the Mie scattering coefficients,$|{B_3}|$ , versus incident wavelength for Helmholtz cavities with radius r = 60 mm and angular width of the opening$\theta = {10^\circ}$ at different incident angle; The E field distribution when an EM wave with$\lambda = 59.2\;{\rm{mm}}$ strikes a Helmholtz cavity with radius r = 60 mm and angle width of the opening$\theta = 10^\circ $ at an angle of incidence c)${\theta _i} = {15^\circ}$ and d)${\theta _i} = {25^\circ}$ 
图 7 a) 半径
$r = 60$ mm, 壁厚$t = 0.2$ mm, 开口角宽度$\theta = {\rm{8}}^\circ $ 的亥姆霍兹腔, 在入射波长$\lambda = 53.8\;{\rm{mm}}$ 时的电场分布图; b) 开口角宽度变为$\theta = 10^\circ $ , 在入射波长$\lambda = 5{\rm{9}}.{\rm{2\;mm}}$ 时的电场分布图. 入射角均为${\theta _i} = 0^\circ $ Fig. 7 a) The E field distribution for the Helmholtz cavity with radius r = 60 mm, wall thickness
$t = 0.2$ mm and angular width of the opening$\theta = {\rm{8}}^\circ $ at$\lambda = 53.8\;{\rm{mm}}$ ; b) The E field distribution for the Helmholtz cavity with radius r = 60 mm, wall thickness$t = 0.2$ mm and angular width of the opening$\theta = 10^\circ $ at$\lambda = 59.2\;{\rm{mm}}$ . The angle of incidence is${\theta _i} = {0^\circ}$ 
图 8 当入射波为横电极化波时, 半径
$r = 60$ mm: a) 开口角宽度$\theta = {\rm{8}}^\circ $ 的亥姆霍兹腔, 在入射波长$\lambda = 53.{\rm{5\;mm}}$ 时的磁场分布图; b) 开口角宽度变为$\theta = 10^\circ $ , 在入射波长$\lambda = 5{\rm{8}}.{\rm{8\;mm}}$ 时的磁场分布图. 入射角均为${\theta _i} = 0^\circ $ Fig. 8 The H field distribution for the Helmholtz cavity with radius r = 60 mm and angular width of the opening
$\theta = {\rm{8}}^\circ $ for the transverse-magnetic polarized wave with$\lambda = 53.5\;{\rm{mm}}$ ; b) The H field distribution for the Helmholtz cavity with radius r = 60 mm and angular width of the opening$\theta = 10^\circ $ at$\lambda = 58.8\;{\rm{mm}}$ . The angle of incidence is${\theta _i} = {0^\circ}$ -
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