In this paper, we consider the problem of the scattering of in-plane waves at an interface between a homogeneous medium and a metamaterial. The relevant eigenmodes in the two regions are calculated by solving a recently described non self-adjoint eigenvalue problem particularly suited to scattering studies. The method efficiently produces all propagating and evanescent modes consistent with the application of Snell’s law and is applicable to very general scattering problems. In a model composite, we elucidate the emergence of a rich spectrum of eigenvalue degeneracies. These degeneracies appear in both the complex and real domains of the wave-vector. However, since this problem is non self-adjoint, these degeneracies generally represent a coalescing of both the eigenvalues and eigenvectors (exceptional points). Through explicit calculations of Poynting vector, we point out an intriguing phenomenon: there always appears to be an abrupt change in the sign of the refraction angle of the wave on two sides of an exceptional point. Furthermore, the presence of these degeneracies, in some cases, hints at fast changes in the scattered field as the incident angle is changed by small amounts. We calculate these scattered fields through a novel application of the Betti–Rayleigh reciprocity theorem. We present several numerical examples showing a rich scattering spectrum. In one particularly intriguing example, we point out wave behavior which may be related to the phenomenon of resonance trapping. We also show that there exists a deep connection between energy flux conservation and the biorthogonality relationship of the non self-adjoint problem. The proof applies to the general class of scattering problems involving elastic waves (under self-adjoint or non self-adjoint operators).

在本文中，我们考虑了平面波在均匀介质与超材料之间的界面处的散射问题。通过解决最近描述的特别适合散射研究的非自伴特征值问题，可以计算出两个区域中的相关本征模。该方法有效地产生了与斯涅尔定律的应用相一致的所有传播模式和modes逝模式，并且适用于非常普遍的散射问题。在模型组合中，我们阐明了丰富的特征值简并性谱的出现。这些简并性同时出现在波矢的复杂域和实际域中。但是，由于此问题不是自伴的，因此这些简并性通常表示特征值和特征向量（例外点）的合并。通过对Poynting向量的显式计算，我们指出了一个有趣的现象：异常点两侧的波的折射角的符号总是出现突变。此外，在某些情况下，这些简并性的存在暗示了随着入射角的微小变化，散射场中的快速变化。我们通过Betti-Rayleigh互易定理的新颖应用来计算这些散射场。我们提供了几个数值示例，这些示例显示了丰富的散射光谱。在一个特别有趣的例子中，我们指出了可能与共振陷波现象有关的波行为。我们还表明，能量通量守恒与非自伴问题的生物正交关系之间存在着深远的联系。