# Propagation of Elastic Waves in a One-Dimensional High Aspect Ratio Nanoparticle Phononic Crystal
## 1 Introduction  
The propagation of elastic and acoustic waves in phononic crystals has been intensely studied over the last three decades [1; 2]. Similarly to photonic crystals that prohibit the propagation of optical or electromagnetic waves [3], phononic crystal can give rise to complete band gaps in which elastic waves are not allowed. The existence and the wideness of the phononic band gaps strongly depend on both the nature of the constituting media and the geometry of the crystal, including the filling fraction and the shape of the inclusions. Complete phononic band gaps have been first demonstrated for bulk waves [2; 4] before being studied for Lamb [5; 6] and surface acoustic waves [7; 8]. Numerous attractive properties of phononic crystals as phononic waveguides, resonators, and stubs have been pointed out over different lengths and frequency scales [9].  
Several works have been devoted to the study of surface acoustic waves (SAW). In contrast to uniform materials where only Rayleigh waves propagate at the surface, additional surface localized vibration modes can be observed when a periodic lattice is deposited on the substrate surface. Simple one-dimensional (1D) periodical ridges built on a uniform substrate were already an issue in the technological application of surface acoustic waves in the interdigitated transducers (IDT) [10]. Recently, they were met with renewed interest thanks to the boost of phononic crystal studies. The 1D periodic grating on surfaces has been first investigated theoretically, considering the propagation of sagittal and shear horizontal acoustic waves [11; 12]. Experimentally, the investigation of the surfaceacoustic band in a 1D periodic array of nanowires has been performed using the laser-induced transient grating technique [13; 14; 15], revealing the resonant modes induced by the nanowires. Brillouin Light Scattering (BLS) has also proven its efficiency for demonstrating the elastic properties of 1D phononic crystal grating [16; 17; 18].  
Due to progress in microfabrication technologies such as microlithography and molecular beam epitaxy, it is now possible to create well defined 1D micro or nanogratings on an otherwise planar surface in which the polymer material presents a high aspect ratio of up to 10 [19]. Additionally, while most of the 1D grating structures are based on semiconductor materials, hybrid 1D gratings comprising hard inorganic and soft polymers represent an alternative flexible platform with distinct advantages: large and tunable band gaps [20; 21; 22], easy fabrication [23], and variation of total thickness and spacing for concurrent operation at hypersonic phonon and visible photon frequencies. The structure investigated here is a one-dimensional nanoridge phononic crystal of finite height grafted on a semi-infinite substrate. The fabrication of such a sample, which is made of a grating of epoxy nanoridges deposited on a glass substrate, has been presented by [PERSON] et al. [23]. Recently, we have shown that the band structure of such a high aspect ratio (AR) nanoridges structure can be partly captured by means of Brillouin light scattering (BLS) experiments [24]. In particular, we demonstrated that the BLS modes can be explained theoretically with a good agreement with their intensities. In the present paper, our main purpose is to deeply investigate the high AR nanoridge structure as a function of the geometrical parameters, considering the two directions of propagation, i.e., parallel and perpendicular to the nanoridges, including wave localization and polarization (Section 3). The second objective, which will be developed in Section 4, is to study the new features in the band structure when the grooves of the native phononic crystal are filled with a liquid.