Analysis of interaction between two SAW modes in Pt grating on langasite cut (0°, 138.5°, 26.6°)

The numerical technique based on a previously developed rational approximation of harmonic admittance of a periodic grating was applied to analysis of SAW behavior in platinum grating on langasite cut with Euler angles (0°, 138.5°, 26.6°). The approximation is able to take into account interaction between surface and bulk waves or between two SAW modes. SAW dispersion was calculated at different values of electrode thickness varying between 1% and 4% of wavelength. It was found that with increasing Pt thickness, SAW behavior in the grating is strongly affected by interaction between two SAW modes propagating in the same orientation. An additional stopband, which results from this interaction, occurs at certain detuning from synchronous reflection condition and can cause spurious resonances of the admittance function. Interaction between two SAW modes is also responsible for anomalously slow growth of reflectivity with increasing platinum thickness.     

Analysis of SAW propagation in gratings on ZnO/diamond substrates

The space harmonic method is used to analyze surface acoustic wave (SAW) propagation under an infinite periodic metal grating on ZnO/diamond composite layered substrates. Dispersion properties for shorted and open gratings are derived as a function of the thickness of the grating electrodes. From these dispersion relations, the coupling of modes (COM) parameters are derived. Energy profiles inside ZnO/diamond show that the energies contained in each of the ZnO and diamond layers are of the same order when the thickness of the ZnO layer is P/pi (P = grating period) and that the energy is contained within two wavelengths below the ZnO/diamond interface.     

Characteristic parameters of surface acoustic waves in a periodic metal grating on a piezoelectric substrate

The coupling problem between electrodes due to the propagation of acoustic waves in a periodic metal grating on a piezoelectric substrate is theoretically studied. A method for the determination of the mutual admittance between the grating electrodes is presented, and an analytical expression for the contribution of surface acoustic waves (SAWs) is proposed. The SAW characteristic parameters are determined with a numerical technique that is able to deal with a leaky SAW as well as a true SAW (conventional Rayleigh type or Bleustein-Gulyaev type). Using this technique, the SAWs' contribution to the mutual admittance can be removed, and the analysis of other contributions becomes possible. In particular, the amplitude decay rate of the residual mutual admittance with electrode separation gives information about the propagation of the surface skimming bulk waves (SSBWs). The method presented is applied to several currently used material-cut configurations.     

Fast,precise, and full extraction of the COM parameters for multielectrode-type gratings by periodic Green's function method

It is very important to extract all four coupling-of-modes (COM) parameters of the electrode cells for the simulation and optimal design of a low loss surface acoustic wave (SAW) filter. A new approach for fast and full extraction of the COM parameters for a multielectrode-type grating is proposed. The field distribution of the wave under the periodic shorted grating is calculated by the periodic Green's function method. The phase of the reflection is determined from the positions of the standing wave node. The transduction coefficient and its phase are determined by the charge distribution at low frequency. The COM parameters of the commonly used electrode width controlled (EWC) single phase unidirectional transducer (SPUDT) is computed. It shows that this is a simple and direct way to extract all the COM parameters for SPUDT and, accordingly, is a powerful tool for the optimization of the filter structure.     

Interface acoustic waves properties in some common crystal cuts

Interface acoustic waves (IAWs), also termed boundary waves, propagate at the interface between two solids. We present two IAW numerical analysis tools, inspired from well established surface acoustic wave (SAW) methods. First, the interface effective permittivity is derived for arbitrary piezoelectric solids and is used to estimate some basic parameters of IAWs. The harmonic admittance for an interface excitation is then derived from the interface effective permittivity, in much the same way the harmonic admittance for surface excitation is obtained from the (surface) effective permittivity. The finite electrode thickness is neglected in this problem analysis. The harmonic admittance is used to model propagation in the case when an infinite periodic interdigital transducer is located at the interface. Simulation results are commented upon for some usual piezoelectric material cuts and the paper outlines a modal selection specific to IAWs as compared with SAWs. The temperature dependence of the resonance frequency is also estimated.     

Stopband characteristics of shallow gratings

A numerical method of rapidly computing the width of the stopband of a shallow grating in a SAW filter and the dependence of this bandwidth on fabrication parameters is presented. The method is based on the properties of the equivalent circuit of the unit cell at the transition frequencies. Numerical results illustrate the variation of the relative bandwidth with respect to the element reflection coefficient, the mark-to-space ratio, and the velocity perturbation parameter.     

Harmonic wave propagation in viscoelastic heterogeneous materials Part I: Dispersion and damping relations

An analytico-numerical method is presented to study the propagation of plane harmonic waves in infinite periodic linear viscoelastic media. Part I considers only the dispersion and attenuation of acoustical longitudinal and shear waves. To show the accuracy of the method, examples of plane harmonic wave propagation in an infinite homogeneous medium and in a periodic layered viscoelastic medium are presented. The method is then used to calculate the damping and dispersion relations for a fibre-reinforced viscoelastic composite material. The results show clearly the influence of materials' viscoelastic properties and heterogeneities on the propagation of plane harmonic waves through the media.     

A treatment of the general volume holographic grating as an array of parallel stacked mirrors

An alternative model to Kogelnik's coupled wave theory of the volume holographic grating is developed in terms of an infinite array of parallel stacked mirrors. The model is based on summing the individual Fresnel reflections from an infinite number of infinitesimal discontinuities in the permittivity profile. The resulting first-order coupled partial differential equations are solved in a rotated frame of reference in order to derive analytical expressions for the diffraction efficiency of the general slanted grating at an arbitrary angle of incidence. The model has been tested using computational solutions of the Helmholtz equation for the unslanted reflection grating. For index modulations characteristic of modern silver halide and photopolymer materials used in display and optical element holography the new model shows excellent agreement with the numerical results. Kogelnik's model also provides good agreement as long as the dephasing parameter is not too large. The model has been tested against Kogelnik's theory for a variety of cases with finite fringe slant with good agreement for typical index modulations. A further advantage of the new model is that colour holographic gratings may be treated at and away from Bragg resonance. Numerical and analytical results are presented concerning the diffractive efficiency of two- and three-colour holographic gratings.     

Finite element computation of Green's functions

Green's functions are important mathematical tools in mechanics and in other parts of physics. For instance, the boundary element method needs to know the Green's function of the problem to compute its numerical solution. However, Green's functions are only known in a limited number of cases, often under the form of complex analytical expressions. In this article, a new method is proposed to calculate Green's functions for any linear homogeneous medium from a simple finite element model. The method relies on the theory of wave propagation in periodic media and requires the knowledge of the finite element dynamic stiffness matrix of only one period. Several examples are given to check the accuracy and the efficiency of the proposed numerical Green's function.     

Scattering of SAW at discontinuities: some numerical experiments

In developing models to characterize the metallic-finger step discontinuity in SAW devices on piezoelectric substrates it is important to estimate the SAW reflection coefficients and the effect of other modes at step discontinuities. In the literature perturbation approximations, finite element methods and various phenomenological parameter-fitting using experimental data are among the techniques which have been exploited. In this paper, a nonperturbation approximation method, which attempts to parallel some well-established techniques used in solving electromagnetic waveguide-discontinuity problems, is explored as an alternate numerical procedure for calculating SAW reflections. Implementing such a method requires that the full solution to SAW propagation in the free and in the layered regions be obtained. Thus, for a free-to-metallized discontinuity, the two SAW phase velocities, the total SAW power now in the free region and in the layered region, and the electroacoustic fields, must be calculated. In this paper the reflection problem is formulated as an optimization problem subject to the necessary scattering parameter constraints. Numerical experiments are described, and the results of calculations for reflectivity are compared to perturbation approximations, nonperturbation approximations and the available experimental data for the effects of energy storage     

Conductance measurements on a leaky SAW harmonic one-port resonator

Conductance measurements are reported on a leaky SAW (LSAW) harmonic one-port resonator on a 64 degrees Y-X LiNbO(3) substrate. This employed a short three-finger IDT for fundamental and second harmonic operation together with long reflection gratings. Conductances were measured with and without the end gratings. From an analysis of the measurements, it was deduced that, for optimum second harmonic performance, the grating stop-band frequency should be higher than the IDT unperturbed center frequency. This result is in contrast to fundamental frequency resonator designs in which the end grating stop-band frequency is placed below the IDT center frequency for optimum performance.     

Infinite boundary elements for dynamic problems of 3-D half space

In this paper, a new procedure for solving 3-D dynamic problems of unbounded foundations in the frequency domain by using BEM is studied. For simulations of wave propagations due to far field effects, a type of infinite boundary element (IBEM) is presented for modelling a 3-D regular or irregular half space. The wave type considered could be compressional, shear or a combination of the two. Through the analysis of the asymptotic behaviour of 3-D fundamental solutions for elasto dynamics, a rather feasible technique for obtaining singular integral coefficients for dynamic problems has been developed. Through the analysis of the dynamic response for a 3-D square foundation under a uniform load distribution, excellent accuracy has been achieved in agreement with previous numerical solutions. Another example–analysis of the dynamic compliance of a rigid square plate on a half space–has also shown very good results. The development of this infinite boundary element provides a powerful tool for dealing with 3-D structure foundation interaction or wave propagation problems for irregular foundations such as arch dam canyons.     

Analysis of leaky-surface-wave propagating under periodic metal grating

A detailed field analysis is presented for a leaky surface wave propagating under a periodic metal grating, using a theory that neglects the effect of mass loading due to the grating. The approach is based on Floquet's theorem and the coupled equations of wave motion with unperturbed mechanical and perturbed (or periodic) electrical boundary conditions, yielding a general field solution applicable to any material and to arbitrary connections to the grating. As a key step, the periodic boundary equations are solved by combining them into a set of infinite homogeneous equations through algebraic treatment and performing orthogonal integration with respect to space harmonics. The advantage in using this method results from there being no need to use assumptions or complicated expressions anticipating an accurate solution if sufficient space harmonics are considered. It is shown that the theory proposed here can be directly extended to solve simpler SAW problems. An analysis is carried out for LiNbO(3) for both the leaky wave and Rayleigh wave, taking into account dispersion relations, propagation attenuation of the leaky wave, and other field distributions. Theoretical and experimental results for the width of the first stopband are discussed.     

The Diffraction Efficiency of Double-grating Holographic Fan-out Elements

Abstract

An analytical expression is derived for the diffraction efficiency of a holographic fan-out element containing two superimposed gratings separated by a small angle. In addition to the usual zero and first orders an infinite set of the significant spurious waves is considered. In deriving the expression all these spurious waves are assumed to satisfy the Bragg condition exactly. The results are compared with a direct numerical solution and give very good agreement for angles up to 1°. Furthermore the analysis provides useful information for angles up to 5°. An analogy is drawn with a hybrid hologram consisting of a thick ‘carrier’ grating and a thin ‘modulation’ grating. Using this model the diffraction efficiencies can be calculated using the standard grating formulae. For fan-out applications the spurious waves around the first orders are considered the most important. From this analysis it can be seen that they can be made negligibly small but at the cost of a reduction in total diffraction efficiency.     

Propagation of longitudinal leaky surface waves under periodic metal grating structure on lithium tetraborate

The dispersion properties of longitudinal leaky surface waves propagating under the periodic Al strip grating on lithium tetraborate (Li(2)B(4)O(7); LBO) are described theoretically and experimentally for applications of the mode to high frequency SAW devices. A theoretical method developed here is based on Floquet's theorem using space harmonics as an orthogonal function set and real boundary integral equations derived from the method of weighted residuals for a period of each region, i.e., substrate, metal, and free space. The boundary integral equations are solved by using the Galerkin procedure. The periodic strip gratings with both single-electrodes and double-electrodes are investigated, considering the convergency of the numerical computation for the number of the space harmonics. As a result, the propagation loss for shorted gratings was found to be relatively low in the thickness range of the Al strip below about 1% for the single-electrodes and 2% for the double-electrodes, although it greatly increases for a thickness over 2% for the single-electrodes and 3% for the double-electrodes.     

Fast and accurate modeling of waveguide grating couplers

A boundary variation method for the analysis of both infinite periodic and finite aperiodic waveguide grating couplers in two dimensions is introduced. Based on a previously introduced boundary variation method for the analysis of metallic and transmission gratings [J. Opt. Soc. Am. A 10, 2307, 2551 (1993)], a numerical algorithm suitable for waveguide grating couplers is derived. Examples of the analysis of purely periodic grating couplers are given that illustrate the convergence of the scheme. An analysis of the use of the proposed method for focusing waveguide grating couplers is given, and a comparison with a highly accurate spectral collocation method yields excellent agreement and illustrates the attractiveness of the proposed boundary variation method in terms of speed and achievable accuracy.     

A dynamic infinite element for three-dimensional infinite-domain wave problems

A three-dimensional dynamic infinite element which satisfies the following requirements: (1) displacement compatibility on the interface between finite and infinite elements; (2) definition of the wave propagation and amplitude attenuation behaviours in the infinite element using wave propagation functions; (3) convergence of the generalized integrals related to mass and stiffness matrices of the infinite element: and (4) displacement continuity along the common boundary of neighbouring infinite elements in the case of simulating multiple material layers or multiple wave numbers within the foundation, is presented in this paper. Since P-waves, S-waves and R-waves in the foundation can be simulated Simultaneously in the present infinite element, the seismic response of an arch-dam-foundation system, especially a thin double-curvature arch-dam-foundation system where the boundary element loses its competitive capacity with the finite element, can be economically calculated by coupling this infinite element with conventional finite elements. The good accuracy obtained using the present infinite element and finite element coupling model to simulate foundation wave problems has been proven by comparing the current numerical results with previous analytical results.     

Spatial Sensitivity Distribution of Surface Acoustic Wave Resonator Sensors

The sensitivity distribution of surface acoustic wave (SAW) resonator sensors is investigated by theoretical and experimental means. It is shown that the sensitivity to mass loading varies strongly across the surface due to the confinement of acoustic energy toward the center of the device. A model is developed for this phenomenon based on the extraction of coupling of modes parameters from a rigorous boundary element method analysis based on a periodic Green's function. As SAW sensors for many applications include a layer covering the electrodes, a new technique is introduced to account for the mechanical interactions with buried electrodes. Using this technique, the sensitivity calculations are found to be in good agreement with measurements. It is also shown that while changes in other parameters influence sensitivity, it is velocity change that most strongly determines overall frequency change     

A network model for arbitrarily oriented IDT structures

A network model approach for analyzing arbitrarily oriented short-circuited SAW grating structures is extended to include interdigital transducers (IDTs) that are also arbitrarily oriented. The IDT structure is divided into cells, each modelled by a sequence of mismatched transmission lines consisting of a metallized and unmetallized region. The model includes: the impedance difference between metallized and free regions, the reflection coefficient at the metallization upstep, the reflection coefficient at the downstep for a counterpropagating wave, all deduced from the Datta-Hunsinger perturbation formula; the velocity difference between the free and metallized regions obtained using SAW propagation calculation software for arbitrarily oriented multilayers; and the energy storage susceptance at each finger discontinuity. Since only ordinary network elements are combined in accordance with the IDT geometry, this model permits good physical insight into the structure's characteristics and allows simple procedures for finding high directivity orientations.     

波动谱元模拟中透射边界稳定性分析

无限域波动数值模拟中,人工边界的稳定性是获得可靠模拟结果的前提。具有高阶精度的谱元法和透射边界两者结合的数值模拟方案显示出较好的模拟精度和数值稳定性,然而,仍然存在数值失稳现象,其失稳机理和稳定条件尚不明确,相应的理论分析极为欠缺。该文针对透射边界在高阶谱元法中的稳定性,依据高阶谱单元中非等间距节点的周期延拓特点,通过构建内域和边界数值格式的向量形式来分析人工边界反射系数。进而保证边界对谱元法中存在的真实模态和虚假模态的反射系数均小于等于1,从而得到透射边界的稳定条件,其表现为无量纲边界参数和谱元参数之间的关系,其含义为透射边界人工波速与介质物理波速的比值限定在一定范围内。同时揭示了透射边界引发高频失稳的机理,即边界对谱元法中虚假模态的反复反射放大所致。最后采用数值实验验证了透射边界稳定条件。