Confined vibrational normal modes in graded elastic networks: Gradons

We discuss the possibility of wave manipulation by graded materials and/or systems. In contrast to the two extreme cases of inhomogeneous system, i.e., periodically modulated system and randomly disordered system, graded system demonstrates a unique way to control wave, resulting in a new type of localization-delocalization transition, which can confine certain vibrational excitations, and redistribute them spatially. We report in detail vibrational normal modes sustained by one-dimensional graded force constant and graded mass networks, in particular, an unusual kind of modes called gradons. By examining the mode characteristics, we put forth an approach to examine the excitation modes in a graded elastic system with the help of a series of homogeneous systems. By this approach, we elaborate the features of the elastic gradons and the phonon-gradon transition. As very useful results, we analytically present the explicit density of states of linearly graded networks in one dimension.