Study On Anisotropic Transmission And Topological Properties Of One Dimensional Local Resonant Phononic Crystals

Phononic crystal is a kind of synthetic material or structure with elastic constant and density distributed periodically,and has elastic band gap.The concept of phononic crystal has attracted people's attention since it was put forward,and it is expected to play a unique role in the manipulation of acoustic or elastic waves.In the past,people mostly focused on the Bragg scattering wavelength range when studying the properties of phononic crystals,but the wavelength range at this time is similar to the structure size,which limits the application scene.Because the wavelength range corresponding to the band gap of local resonance phononic crystals can be more than one order of magnitude larger than its size,it has more application potential.In 1998,Bender and Boettcher studied the relationship between the parity time(PT)symmetry and the eigenspectrum of non Hermite Hamiltonian operators.They pointed out that the eigenvalues of non-Hermite Hamiltonian operators can also be real if they satisfy the PT-symmetry,and further found that the breaking of the parity time symmetry of nonHermite Hamiltonian operators will lead to the eigenvalues changing from real to complex.Recently,researchers have found that PT-symmetry non-Hermite system could also be achieved in classical wave system due to the similarity between the time-varying Schrodinger equation and paraxial diffraction equation.The research on topology is first started in condensed matter physics.Recently,acoustic and elastic waves have been proved to be a well platform for studying and observing topological phenomena.Through these topological phenomena,many unique functions can be realized,such as backscatter suppression,defect of immune,acoustic logic gate,etc.In this paper,we first study the properties of PT symmetric non-Hermitian one-dimensional local resonant phononic crystal,and then study their topological properties under Hermite condition.The details are as follows:(1)A core cladding theoretical model is designed.By analyzing the force on each part of the structure,its dynamic equations are established,then we solve its secular determinant to obtain the energy band dispersion relation.After that,by introducing the positive imaginary part and negative imaginary part with equal absolute value,which are equivalent to introducing loss and gain,a non-Hermitian system satisfying Pt symmetry is formed.At this time,the energy band degenerates at the boundary of Brillouin zone and produces exceptional points.If the intensity of loss and gain is increased,the local resonance band gap will disappear,different exceptional points will move and finally merge,resulting in new exceptional point.(2)Compared with the theoretical model,the practical structure was constructed with epoxy as the core,rubber as the cladding and water as the substrate.The energy band of the actual structure is obtained by using the finite element simulation software,and compared with the energy band calculated by the theoretical model,which proves the rationality of the practical structure corresponding to the theoretical model.By calculating the transmission spectrum of its finite structure,we verify that it has anisotropic transmission resonance phenomenon in the subwavelength region.(3)It is found that the degeneracy point caused by band folding can be opened by changing the distance between scatterers in the cell,resulting in band gap and isolated band.Then,according to the zak phase of these isolated bands,topological phase transition is found in the subwavelength frequency range.Our work successfully combines the local resonant phononic crystal with the nonHermite physics,which enriches the physical connotation of the non Hermite local resonant phononic crystal.