Second-harmonic generation in an infinite layered structure with nonlinear spring-type interfaces

Abstract

The second-harmonic generation characteristics in the elastic wave propagation across an infinite layered structure consisting of identical linear elastic layers and nonlinear spring-type interlayer interfaces are analyzed theoretically. The interlayer interfaces are assumed to have identical linear interfacial stiffness but can have different quadratic nonlinearity parameters. Using a perturbation approach and the transfer-matrix method, an explicit analytical expression is derived for the second-harmonic amplitude when the layered structure is impinged by a monochromatic fundamental wave. The analysis shows that the second-harmonic generation behavior depends significantly on the fundamental frequency reflecting the band structure of the layered structure. Two special cases are discussed in order to demonstrate such dependence, i.e., the second-harmonic generation by a single nonlinear interface as well as by multiple consecutive nonlinear interfaces. In particular, when the second-harmonic generation occurs at multiple consecutive nonlinear interfaces, the cumulative growth of the second-harmonic amplitude with distance is only expected in certain frequency ranges where the fundamental as well as the double frequencies belong to the pass bands of the layered structure. Furthermore, a remarkable increase of the second-harmonic amplitude is found near the terminating edge of pass bands. Approximate expressions for the low-frequency range are also obtained, which show the cumulative growth of the second-harmonic amplitude with quadratic frequency dependence.

Second-harmonic generation in a multilayered structure with nonlinear spring-type interfaces embedded between two semi-infinite media

“…When all interlayer interfaces possess the same nonlinearity of β m = β (m = 1, 2, …, N-1), the second-harmonic amplitudes on both sides of the multilayered structure are calculated by Eqs. (52) and (53). Their variation with the normalized fundamental frequency is shown in Fig.…”

“…, where the stress is continuous while the discontinuity is allowed in the displacement. When the spring-type interfaces possess weak quadratic nonlinearity [43], the boundary conditions for the mth interface are given by [52], [53] , ,…”

“…By introducing the Bloch wavenumber for the infinitely extended layered structure whose unit-cell consists of one layer and one linearized spring-type interface given by [31], [52] exp…”

“…For the more general case of multiple interfaces, Biwa and Ishii [52] analyzed the propagation characteristics of the Bloch wave in infinitely layered structures with spring-type interlayer interfaces possessing a weak quadratic nonlinearity. They elucidated the frequency dependence of the second-harmonic generation in the structures based on their pass and stop band characteristics.…”

“…Following the perturbation analysis for a weak quadratic nonlinearity of interfaces carried out in Ref. [52], the governing equations for the propagation of fundamental wave and its second-harmonic component in the structure embedded between two linear elastic semi-infinite media are presented in the frequency domain in Section 2. The solution to the fundamental wave propagation, i.e., the amplitude reflection and transmission coefficients as well as the displacement jumps at the interlayer interfaces, is obtained using the transfer-matrix method in Section 3.…”