发表机构
École Polytechnique（巴黎综合理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过体积-势矩阵约化，证明单调材料分级可使色散谐振器阵列实现解析可控的频率排序，并给出逆设计方法及数值验证。
AI 中文摘要
我们通过体积-势矩阵约化研究高度色散的亚波长谐振器的分级阵列。该约化产生一个有限维非线性共振问题，其对角元素编码局部色散响应，非对角元素编码不同谐振器之间的体积介导耦合。我们证明，单调材料分级将该矩阵转化为解析可控的频率排序器。当分级失谐在约化特征矩阵中主导耦合时，特征向量局域化在单个谐振器附近；在显式的半晶格位点局域化条件下，这种局域化变为单调的频率-位置映射。我们还证明了一个建设性的逆设计命题：通过选择局部色散参数可以实现指定的单调目标频率，并且对足够小的分级误差具有鲁棒性。数值实验（其参数和算法记录在附录中）展示了局域共振、设计的频率-位置映射、无序稳定性、间距依赖性，以及超越前导对数求积模型的已解析的物理Lippmann-Schwinger验证。
英文摘要
We study graded arrays of highly dispersive subwavelength resonators through their volume-potential matrix reduction. The reduction gives a finite-dimensional nonlinear resonance problem whose diagonal entries encode the local dispersive response and whose off-diagonal entries encode volume-mediated coupling between distinct resonators. We show that monotone material grading turns this matrix into an analytically controllable frequency sorter. When graded detuning dominates coupling in the reduced characteristic matrix, the characteristic vectors localize near individual resonators; under an explicit half-lattice-site localization condition, this localization becomes a monotone frequency-position map. We also prove a constructive inverse-design statement: prescribed monotone target frequencies can be realized by choosing the local dispersive parameters, with robustness under sufficiently small grading errors. Numerical experiments, whose parameters and algorithms are recorded in an appendix, illustrate localized resonances, designed frequency-position maps, disorder stability, spacing dependence, and a resolved physical Lippmann-Schwinger validation beyond the leading logarithmic quadrature model.
Comments25 pages, 5 figures
