1.Key Laboratory of Luminescence and Optical Information, Ministry of Education, Institute of Optical Information, School of Science, Beijing Jiaotong University, Beijing 100044, China 2.State Key Laboratory on Integrated Optoelectronics, Beijing 100083, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant No. 61571035) and the State Key Laboratory on Integrated Optoelectronics, China (Grant No. IOSKL2018KF22).
Received Date:21 January 2019
Accepted Date:28 March 2019
Available Online:01 August 2019
Published Online:05 August 2019
Abstract:A C-band rectangular waveguide with gyroelectric semiconductor is designed to study the non-reciprocal propagation characteristics of surface magnetoplasmons (SMPs), which are generated by an external magnetic field. The effective refractive index method is used to obtain the effective refractive index and transverse electric field distribution of the waveguide, and a two-dimensional rectangular waveguide is approximately regarded as a combination of two one-dimensional planar waveguides. The dispersion equation of planar waveguide with $ {\rm{E}}_{mn}^x$ mode in rectangular waveguide is derived. The influences of the structural parameters of rectangular waveguide and the refractive index of material on the non-reciprocal dispersion relation and time-delay characteristics are analyzed by numerical method. Due to the effect of external magnetic field, the off-diagonal elements of dielectric tensor in magnetic photonic crystal are changed. The generation of electrical anisotropy leads the time reversal symmetry to be broken. As a result, the dispersion curves of the rectangular waveguide are asymmetric with respect to propagation constant, and the complete one-way transmission of SMPs can be realized in the asymmetric frequency region. The dispersion curve tends to be a dispersion curve of planar waveguide as the width of rectangular waveguide increases, but the non-reciprocal frequency range is approximately unchanged. The width of the core region and the refractive index of the side material have a significant influence on the non-reciprocal dispersion characteristics: the group velocity of SMPs decreases with ω and propagation constant decreasing. The group velocity is related to the waveguide width, propagation constant and the operating wavelength. The relationship between the normalized group velocity and the width of the waveguide separately operating at 1530, 1550 and 1565 nm are studied. The group velocity is relatively slow when the width of waveguide’s core region is between 140 nm and 233.5 nm, and the minimum group velocity reaches 5.43 × 10-2c. As for the slow light effect, the rectangular waveguide is better than planar waveguide. The rectangular waveguide has a large engineering tolerance in the width of core region, which is 93.5 nm. In addition, the dispersion curves of the rectangular waveguide with SiO2, Air, Au and Ag as the left and right cladding layers are calculated. As a result, the group velocity is proportional to the refractive index of the side material in the y direction of the rectangular waveguide. The slow light effect is the most obvious when the material is silver, and the minimum transmission speed can reach 2.8 × 10-3c. Keywords:rectangular waveguide/ effective refractive index/ surface magnetoplasmons/ nonreciprocal properties
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2.1.波导结构与旋电材料介电张量
光通信C波段旋电材料的矩形波导结构的截面如图1所示, 该波导由电介质层(Si)、光通信C波段旋电半导体层(gyroelectric semiconductor, GS)以及四面有界的外包层(Ag)组成. 其中波导芯区的y方向和Si层x方向的宽度分别为2a和d, 选取Si和GS交界面的中心为原点建立二维直角坐标系, z轴垂直于纸面向里. 在–y方向上对旋电半导体层施加静磁场B0, 由于磁场对介电张量的影响[2]使得SMPs在旋电材料矩形波导中沿z轴正方向非互易传输, 打破了该波导系统的洛伦兹互易性[20]. 图 1 基于光通信C波段旋电材料的矩形波导结构 Figure1. Structure of rectangular waveguide with gyroelectric material in C-band.
利用有效折射率法, 把一个二维矩形波导近似看成两个一维平面波导(planar waveguide, PW)的组合, 即x方向受约束的平面波导PW1和y方向受约束的平面波导PW2, 分别见图2(a)和图2(b). 图 2 有效折射率法的两个等效平面波导截面图 (a) x方向受约束的平面波导PW1; (b) y方向受约束的平面波导PW2 Figure2. Sectional views of two equivalent planar waveguides by effective refractive index method: (a) Planar waveguide PW1 with x direction constraint; (b) planar waveguide PW2 with y direction constraint.