fig3a_data.txt from Predicting double negativity using transmitted phase in space coiling metamaterials

Metamaterials are engineered materials that offer the flexibility to manipulate the incident waves leading to exotic applications such as cloaking, extraordinary transmission, sub-wavelength imaging and negative refraction. These concepts have largely been explored in the context of electromagnetic waves. Acoustic metamaterials, similar to their optical counterparts, demonstrate anomalous effective elastic properties. Recent developments have shown that coiling up the propagation path of acoustic wave results in effective elastic response of the metamaterial beyond the natural response of its constitutive materials. The effective response of metamaterials is generally evaluated using the ‘S’ parameter retrieval method based on amplitude of the waves. The phase for acoustic waves contains information of wave pressure and particle velocity. Here, we show using finite-element methods that phase reversal of transmitted waves may be used to predict extreme acoustic properties in space coiling metamaterials. This change in phase is the difference in the phase of the transmitted wave with respect to the incident wave. This method is simpler when compared with more rigorous, the ‘S’ parameter retrieval method. The inferences drawn using this method have been verified experimentally for labyrinthine metamaterials by showing negative refraction for the predicted band of frequencies.

超材料（Metamaterials）是一类人工设计的功能材料，可灵活调控入射波，进而实现隐身、异常透射、亚波长成像以及负折射等极具颠覆性的应用。这类概念目前多在电磁波领域得到深入探索。声学超材料（Acoustic metamaterials）与光学超材料同类相仿，可展现出反常的有效弹性特性。近期研究表明，通过卷曲声波的传播路径，声学超材料可获得远超其组成材料本征响应的有效弹性性能。超材料的有效响应通常采用基于波幅的S参数反演法（S parameter retrieval method）进行评估。声波的相位蕴含了声压与质点振动速度的相关信息。本文借助有限元法（finite-element methods）证明，透射波的相位反转可用于预测空间卷曲型声学超材料的极端声学特性。此处的相位变化指透射波相对于入射波的相位差。相较于更为严谨的S参数反演法，该方法更为简便。针对迷宫型声学超材料（labyrinthine metamaterials），本文通过在预测频段内观测到负折射现象，从实验层面验证了该方法推导结论的可靠性。