Extending completeness of the eigenmodes of an open system beyond its boundary, for Green’s function and scattering-matrix calculations: data

In the related research the asymptotic completeness of eigenmodes is investigated. The asymptotic completeness of a set of the eigenmodes of an open system with increasing number of modes enables an accurate calculation of the system response in terms of these modes. Using the exact eigenmodes, such completeness is limited to the interior of the system. In paper it is shown that when the eigenmodes of a target system are obtained by the resonant-state expansion, using the modes of a basis system embedding the target system, the completeness extends beyond the boundary of the target system. This is illustrated by using the Mittag-Leffler series of the Green’s function expressed in terms of the eigenmodes, which converges to the correct solution anywhere within the basis system, including the space outside the target system. Importantly, this property allows one to treat pertubations outside the target system and to calculate the scattering crosssection using the boundary conditions for the basis system. Choosing a basis system of spherical geometry, these boundary conditions have simple analytical expressions, allowing for an efficient calculation of the response of the target system, as demonstrated for a resonator in a form of a finite dielectric cylinder.Data is given separately for each figure in the paper: in .opju format, which can be viewed by the free software OriginViewer, and contains the figures from the paper; and in .xls format, which contains the raw data only.FIG. 1. a) RSE modes (crosses) of a dielectric sphere with ε = 9, basis radius R, and target radius 0.7R, in the complex wave number plane, along with the exact modes of the basis (dots) and target system (squares). b) Field amplitude of a mode close to a physical RS (green) and a lower and a higher order VG mode (blue, black), with arrows in a) indicating these modes for N = 39.FIG. 2. a) Comparison of the exact analytic form of the GF (Gan) and its ML series using exact modes (GML) with N = 39 (kmaxR ≈ 28.5), and modes calculated via the RSE (GRSE) with N = 39 (kmaxR ≈ 20), for a source located in the gap at r′ = 0.85R (vertical dashed line), for kR = 5. b) Relative error of the ML series of the GF, with one point and both points on the surface of the basis sphere, as labelled, for kR = 5, calculated via Eq. (4) (solid) and Eq. (3) (dashed).FIG. 3. a) Complex k-plane with modes of a target dielectric sphere (ε = 9) calculated with the RSE for l = 1, 2, both in TE and TM polarizations, with a basis kmaxR ≈ 20 (N = 158). b) Scattering cross-section of a perturbed sphere of radius Rp = 0.7R, with basis modes from a), calculated on the target (dashed lines) and basis surface (solid lines), with (teal) and without (red) VG modes, and the black line showing the exact solution.FIG. 4. Scattering cross-section of a cylinder, with ε = 9, height and diameter of sqrt(2)R, and incoming excitation propagating along the cylinder axis (ki), calculated with kmaxR = 13 (N = 1004). The eigenmodes in the complex plane are shown for comparison (crosses, right axis).Research results based upon these data are published at https://doi.org/10.1103/PhysRevResearch.7.L012035

本研究围绕开放系统本征模的渐近完备性展开探讨。当开放系统的本征模数量不断增加时，其本征模集合所具备的渐近完备性，可使研究者借助这些模精准计算系统的响应特性。若采用精确本征模，这类完备性仅局限于系统内部区域。已有研究表明，当通过共振态展开（Resonant-State Expansion, RSE），利用嵌入目标系统的基系统的本征模来获取目标系统的本征模时，其完备性可延伸至目标系统边界之外。这一结论可通过以本征模形式表示的格林函数（Green’s function）米塔-列夫勒（Mittag-Leffler）级数得到验证：该级数在基系统的全域（包括目标系统外部空间）内均可收敛至正确解。值得注意的是，这一特性允许研究者对目标系统外部的微扰进行处理，并可基于基系统的边界条件计算散射截面。若选择球形几何结构作为基系统，其边界条件可简化为简洁的解析表达式，从而实现目标系统响应的高效计算，这一点已针对有限介质圆柱形式的谐振器得到验证。

本文各图对应数据均单独提供：其一为.opju格式文件，可通过免费软件OriginViewer查看，内含论文中的全部插图；其二为.xls格式文件，仅包含原始实验数据。

图1 a) 介电常数ε=9的介质球的共振态展开（RSE）本征模（十字标记），基半径为R，目标半径为0.7R，复波数平面分布，同时标注基系统（圆点）与目标系统（方块）的精确本征模。b) 接近物理共振态（Resonant State, RS）的本征模场振幅（绿色曲线），以及低阶与高阶VG模（蓝色、黑色曲线），图a中的箭头对应N=39时的上述模。

图2 a) 针对位于r′=0.85R间隙处的源（垂直虚线标记位置），比较格林函数（GF）的精确解析形式（Gan）、采用N=39精确本征模的米塔-列夫勒级数解（GML，kmaxR≈28.5），以及采用N=39共振态展开（RSE）计算得到的模的解（GRSE，kmaxR≈20），计算参数为kR=5。b) 针对kR=5的情况，格林函数米塔-列夫勒级数的相对误差：分别以基球表面的单点与双点作为采样点（如图注标注），分别通过式(4)（实线）与式(3)（虚线）计算得到。

图3 a) 复波数平面：针对l=1、2的横电波（Transverse Electric, TE）与横磁波（Transverse Magnetic, TM）偏振模式，采用共振态展开（RSE）计算得到的目标介质球（ε=9）本征模分布，基参数为kmaxR≈20（N=158）。b) 半径Rp=0.7R的微扰球体的散射截面：采用图a中的基系统模，分别在目标表面（虚线）与基表面（实线）计算结果，其中包含VG模的结果为青绿色曲线，未包含VG模的结果为红色曲线，黑色曲线为精确解析解。

图4 介电常数ε=9的圆柱的散射截面，其高度与直径均为√2 R，入射激发沿圆柱轴线（ki）传播，计算参数为kmaxR=13（N=1004）。同时在复平面中展示本征模分布以作对比（十字标记，右侧坐标轴）。

基于本数据集的研究成果已发表于https://doi.org/10.1103/PhysRevResearch.7.L012035