fig3b.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）是一类人工设计的材料，可灵活操控入射波，从而实现隐身（cloaking）、异常透射（extraordinary transmission）、亚波长成像（sub-wavelength imaging）与负折射（negative refraction）等诸多奇异应用。这些概念大多已在电磁波（electromagnetic waves）场景下得到广泛研究。声学超材料（Acoustic metamaterials）与光学超材料同类物类似，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，可使该超材料获得远超其组成材料本征响应的有效弹性响应。超材料的有效响应通常采用基于波幅的“S参数检索法（'S' parameter retrieval method）”进行评估。声波的相位包含波压与质点振动速度的信息。本文采用有限元法（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。该相位变化指的是透射波相对于入射波的相位差。相较于更为严谨的S参数检索法，该方法更为简便。针对迷宫型声学超材料，通过在预测频段内观测到负折射现象，已验证了该方法所得推论的正确性。