Chen Yi, Schneider Jonathan L G, Wang Ke, Scott Philip, Kalt Sebastian, Kadic Muamer, Wegener Martin
Institute of Nanotechnology, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany.
Institute of Applied Physics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany.
Nat Commun. 2024 Oct 24;15(1):8882. doi: 10.1038/s41467-024-52956-5.
Evanescent Bloch waves are eigensolutions of spatially periodic problems for complex-valued wavenumbers at finite frequencies, corresponding to solutions that oscillate in time and space and that exponentially decay in space. Such evanescent waves are ubiquitous in optics, plasmonics, elasticity, and acoustics. In the limit of zero frequency, the wave "freezes" in time. We introduce frozen evanescent waves as the eigensolutions of the Bloch periodic problem at zero eigenfrequency. Elastic waves, i.e., phonons, in metamaterials serve as an example. We show that, in the complex plane, the Cauchy-Riemann equations for analytical functions connect the minima of the phonon band structure to frozen evanescent phonons. Their exponential decay length becomes unusually large if a minimum in the band structure tends to zero and thereby approaches a soft mode. This connection between unusual static and dynamic behaviors allows to engineer large characteristic decay lengths in static elasticity. For finite-size samples, the static solutions for given boundary conditions are linear combinations of frozen evanescent phonons, leading to interference effects. Theory and experiment are in excellent agreement. Anomalous behavior includes the violation of Saint Venant's principle, which means that large decay-length frozen evanescent phonons can potentially be applied in terms of remote mechanical sensing.
倏逝布洛赫波是有限频率下复数值波数的空间周期问题的本征解，对应于在时间和空间中振荡且在空间中指数衰减的解。这种倏逝波在光学、等离子体学、弹性力学和声学中普遍存在。在零频率极限下，波在时间上“冻结”。我们将冻结倏逝波定义为零本征频率下布洛赫周期问题的本征解。超材料中的弹性波，即声子，就是一个例子。我们表明，在复平面中，解析函数的柯西 - 黎曼方程将声子能带结构的最小值与冻结倏逝声子联系起来。如果能带结构中的最小值趋于零并因此接近软模，其指数衰减长度会变得异常大。这种异常的静态和动态行为之间的联系使得在静态弹性中设计大的特征衰减长度成为可能。对于有限尺寸的样品，给定边界条件下的静态解是冻结倏逝声子的线性组合，从而导致干涉效应。理论与实验结果吻合得非常好。异常行为包括违反圣维南原理，这意味着大衰减长度的冻结倏逝声子在远程机械传感方面具有潜在的应用价值。
