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

用点匹配的方法研究高介电常数半球形谐振器中回声廊模的共振特性。1999年，我们用球面波求和的方法来表示谐振模的场分布。在边界上的适当匹配点上，将谐振器内的切向电磁场与谐振器外的切向电磁场匹配。导出了4N×4N(N；匹配点数)行列式作为半球谐振腔的本征值方程。在本分析中，由于N个匹配点位于谐振器的边界上，使其成等角，无法获得具有良好收敛的谐振频率和卸载Q因子。2000年，我们改进了匹配点的位置，得到了收敛良好的特征值。对于匹配点的定位，我们还介绍了一种根据语廊模式的电磁场分布来确定匹配点的新技术。由于关联勒让德函数P^m_n(cosθ)给出了共振模与方位角的依赖关系，因此我们定义横坐标为θ。作为P^m_&lt；m+2Z-1&gt；(cosθ)=0的解。考虑到场的对称性，我们还将本征值方程修改为表示为(4N-2)×(4N-2)行列式的新的本征值方程。从数值分析的结果可以看出，对于匹配点数N，谐振频率和空载Q因子具有很好的收敛特性，数值计算结果与实验结果吻合较好。

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.