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
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.
3.波导宽度对非互易色散特性的影响利用色散方程(16)计算出不同芯区宽度的矩形波导色散曲线, 如图3(a)所示, 波导结构为Ag材料四面包裹电介质层和旋电半导体层, 参数见2.2节. 由于对洛伦兹互易性的破坏, 其色散曲线关于波矢k不对称, 在不对称的频率(图中两红色水平虚线之间)区域内可以实现完全的单向传输, 波导表现出非互易性. 从图3(a)可以看出, 随着矩形波导半宽度a由0.02λp, 0.04λp, 0.06λp, 0.08λp, 0.12λp增大至0.16λp, 其色散曲线自然趋向平面波导的色散曲线(图3(a)蓝色实线), 但是非互易的频率区间基本不变. 图 3 (a)不同芯区宽度的矩形波导色散曲线; (b)不同芯区宽度的矩形波导中SMPs波单向传输区域的群速度; (c)不同波长的SMPs波群速度随芯区宽度的变化 Figure3. (a) Dispersion curves of rectangular waveguide with different core widths; (b) group velocity of one-way SMPs transmission region in rectangular waveguide with different widths; (c) variation of group velocity of SMPs with different wavelengths with different core widths.
为了讨论非互易波导的缓存性能, 图3(b)给出了不同半宽度a的矩形波导, 其SMPs波归一化群速度${v_{\rm{g}}}/c = {\rm{d}}\omega /(c \cdot {\rm{d}}k)$与归一化角频率ω/ωp和归一化传播常数k/kp的关系. 从图3(b)可以看出, 不管是随着ω (虚线)增大还是随着k (实线)增大, 均出现SMPs波群速度减慢现象. 为了研究波导宽度对群速度减慢效应的影响, 图3(c)给出了工作波长λ在1530, 1550和1565 nm处的归一化群速度随波导宽度的变化. 波导半宽度a在0.06λp (波导宽度2a = 140 nm)与0.10λp (2a = 233.5 nm)之间的vg相对较低, 在约0.08λp处达最小值. 而且随λ减小, vg减小, 图中最小群速度达到5.43 × 10–2c. 这一结论表明, 宽度适当的矩形波导的非互易慢光效应比平面波导的非互易慢光效应更明显, 而且在C波段工艺容差较大(约为93.5 nm). 4.折射率对非互易色散特性的影响本文计算了矩形波导左右外包层分别为半导体SiO2、空气Air、金Au、银Ag时的色散曲线, 如图4(a), 材料折射率从0.14, 0.52, 1.00到1.45, 其色散曲线逐渐趋向平面波导色散曲线. 图4(b)是单向传输区域中SMPs波归一化群速度vg/c与归一化角频率ω/ωp和归一化传播常数k/kp的关系, 该图显示出与图3(b)相似的规律, 即SMPs波vg随ω (虚线)或k (实线)的增大而减慢. 随着材料折射率递减, SMPs波群速度逐渐减小, 群时延增大, 慢光效应越明显. 矩形波导左右外包层材料为Ag时SMPs波群速度最小(vg = 2.8 × 10–3c), 慢光效应最显著. 图 4 (a)不同材料的矩形波导色散曲线; (b)不同材料的矩形波导中SMPs波单向传输区域的群速度曲线 Figure4. (a) Dispersion curves of rectangular waveguide with different materials; (b) group velocity of one-way SMPs transmission region in rectangular waveguide with different materials.