A Basic Study on a Resonance Characteristics of a Semi-Spherical Resonator with a High Permittivity in Millimeter Wave Band

本文的研究课题是利用点匹配方法在高介电常数的半球形谐振器中明确窃窃廊模式的谐振特性。1999年，我们用球面波的和表示了共振模态的场分布。将谐振腔内的切向电磁场与谐振腔外的切向电磁场在边界上适当的匹配点进行匹配。得到了半球形谐振器的特征值方程4N×4N （N；匹配点数）行列式。由于本分析中有N个匹配点位于谐振器的边界上，形成等角，因此我们无法获得收敛性较好的谐振频率和卸载Q因子。2000年，我们改进了匹配点的位置，得到了收敛性好的特征值。对于匹配点的位置，我们还引入了一种新的技术，该技术来源于窃窃廊模式的电磁场分布。由于共振模数为m的耳语廊模的方位角依赖由相关的勒让德函数P^m_n （cos θ）表示，因此我们定义了横坐标θ。作为P^m_<m+2Z-1> (cos θ)=0的解。考虑到场的对称性，我们还将特征值方程修改为表示为（4N-2）×（4N-2）行列式的新特征值方程。从数值分析结果可以看出，共振频率和卸载Q因子在匹配点数目n上收敛得很好，数值结果与实验结果比较吻合。

A subject of our research is to make clear resonance characteristics of Whispering Gallery Modes can a semi-spherical resonator with a high permittivity by a Point Matching Method.In 1999, We expressed field distributions of the resonance modes by a summation of spherical waves. Tangential electromagnetic fields inside the resonator were matched to those outside the resonator at appropriate matching points on the boundary. As the result, a 4N×4N (N ; number of matching points) determinant was derived as an eigenvalue equation of the semi-spherical resonator. As N matching points, in this analysis, were located on a boundary of the resonator to make equi-angles, we failed to obtain resonance frequencies and unloaded Q factors with a good convergence. In 2000, we improved our location of the matching point to obtain eigenvalue with a good convergence. For a location of the matching points, we also introduced a new technique derived from electromagnetic field distribution of the Whispering Gallery Modes. Since an azimuthal angle dependence of the Whispering Gallery Mode with a resonance mode number m was presented by the associated Legendre function P^m_n (cos θ), we defined abscissas θ. of the matching points as solutions of P^m_<m+2Z-1> (cos θ)=0. Considering the field symmetry, we also modified the eigenvalue equation to a new eigenvalue equation which was expressed (4N-2)×(4N-2) determinant. From results of numerical analysis, we could find that the resonance frequencies and unloaded Q factor well converged for number of matching points N.A comparison of numerical results and experimental ones showed a good agreement with each other.