We show that topological order and vibrational edge modes can exist in a classical mechanical system consisting of a two-dimensional honeycomb lattice of masses and springs. The band structure shows the existence of Dirac cones and unconventional edge states that are similar to the vibrational modes in graphene. Interestingly, as the system is placed on a constantly rotational coordinate system, the Coriolis force resulting from the non-inertial reference frame introduces time-reversal symmetry breaking and leads to topologically nontrivial band gaps. The nontrivial topological orders are further verified by the calculation of Chern numbers for corresponding bands.

The power loss and electromagnetic energy density of a metamaterial consisting of arrays of wires and split-ring resonators are investigated. We show that a field energy density formula can be derived consistently from both the electrodynamic (ED) approach and the equivalent circuit (EC) approach. The derivations are based on the knowledge of the dynamical equations of the electric and magnetic dipoles in the medium and the correct form of the power loss. We discuss the role of power loss in determining the form of energy density and explain why the power loss should be identified first in the ED derivation. When the power loss is negligible and the field is harmonic, our energy density formula reduces to the result of Landau's classical formula. For the general case with finite power loss, our investigation resolves the apparent contradiction between the previous results derived by the EC and ED approaches.

We derive the electromagnetic energy density in a chiral metamaterial consisting of uncoupled single-resonance helical resonators. Both the lossless and absorptive cases are studied, and the energy density is shown to be positively definite. The key relation making the derivation successful is the proportionality between the magnetization and the rate of change of the electric polarization of the medium. The same time-domain formulation of energy density also applies to the bianisotropic medium proposed by Zhang et al. [Phys. Rev. Lett.102, 023901 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.023901]. This work may provide insights for studying time-dependent phenomena in metamaterials.

Transmission properties of the periodic dielectric waveguide (PDWG) formed by aligning a sequence of dielectric cylinders in air are investigated theoretically. Unlike photonic crystal waveguides (PCWs), light confinement in a PDWG is due to total internal reflection. Besides, the dispersion relation of the guided modes is strongly influenced by the dielectric periodicity along the waveguide. The band structure for the guided modes is calculated using a finite-difference time-domain (FDTD) method. The first band is used for guiding light, which makes PDWG single mode. Transmission is calculated using the multiple scattering method for various S shaped PDWGs, each containing two opposite bends. When PDWG operates in appropriate frequency ranges, high transmission (above 90%) is observed, even if the radius of curvature of the bends is reduced to three wavelengths. This feature indicates that the guiding ability of PDWG can be made better than the conventional waveguide when used in an optical circuit. In addition, PDWG has the advantage that it can be bent to any arbitrary shape while still preserves the high transmission, avoiding the geometric restriction that PCWs are subject to.

Propagation of acoustic waves in an one-dimensional water duct containing many air filled blocks is studied by the transfer matrix formalism. Energy distribution and interface vibration of the air blocks are computed. For periodic arrangement band structure and transmission rate are calculated analytically, whereas the Lyapunov exponent and its variance are computed numerically for random situations. A distinct collective behavior for localized waves is found. The results are also compared with optical situations.