## 1. Introduction

The bulk acoustic wave (BAW) devices first emerged in 1920s and the surface acoustic wave (SAW) devices first appeared in 1960s (Royer & Dieulesaint, 2000). Since invented, these acoustic wave devices have been improved greatly in their performance and applications, along with significantly extended working parameters and application areas (Royer & Dieulesaint, 2000; Hashimoto, 2000). Nevertheless, in the last two decades, even more rigorous demands such as high operational frequency, high sensitivity, high reliability, multiple functionality, broad environment applicability, low attenuation and low cost, arise from the consumer, commercial and military applications. These demands challenge the conventional acoustic wave devices in which single crystalline piezoelectric materials are used as the wave medium. Therefore, the scheme of innovative acoustic wave devices utilizing piezoelectric multi-layered (stratified) structures was presented to cater for these demands. Fortunately, the successes of thin film deposition, etching and lithography technologies lead to the availability of piezoelectric multi-layered structures (Benetti et al., 2005). Recently, high-performance acoustic wave devices with multi-layered structures have been contrived and successfully fabricated (Kirsch et al., 2006; Benetti et al., 2008; Nakanishi et al., 2008; Brizoual et al., 2008). To further reduce the acoustic loss and enhance the quality factor of the multi-layered acoustic wave devices, Bragg Cell composed of many thin periodic alternate high- and low-impedance sublayers can be inserted between the propagation layer and the substrate. Efforts have been made on the fabrication of integrated piezoelectric multi-layered materials with Bragg Cell (Yoon & Park, 2000) and on the realization of superior multi-layered acoustic wave devices with Bragg Cell (Chung et al., 2008), especially aiming at the film bulk acoustic resonators (FBAR).

To ensure the well and stable performance of multi-layered acoustic wave devices, clear understanding of their operation status, especially the acoustic wave propagation behavior, is indispensable in the design process. Therefore, accurate and reliable modeling methods are necessary. By far, three sorts of matrix methods, including the analytical methods based on continuous (distributed-parameter) models, numerical methods based on discrete models and analytical-numerical mixed methods, have been presented for analyzing multi-layered acoustic wave devices. Analytical matrix methods, such as the transfer matrix method (TMM) (Lowe, 1995; Adler, 2000), the effective permittivity matrix method (Wu & Chen, 2002), the scattering matrix method (Pastureaud et al., 2002), and the recursive asymptotic stiffness matrix method (Wang and Rokhlin, 2002), usually give accurate results with low computational cost. However, some of these analytical methods are numerically instable. One reason is that both exponentially growing and decaying terms with respect to frequency and thickness are incorporated in a same matrix, and the other is that matrix inversion is involved in the formulation. For example, TMM ceases to be effective for cases of high frequency-thickness products. Tan (2007) compared most analytical methods in their mathematical algorithm, computational efficiency and numerical stability. Very recently, Guo et al. (Guo, 2008; Guo & Chen, 2008a, 2010; Guo et al., 2009) have presented a new version of the analytical method of reverberation-ray matrix (MRRM) formerly proposed by Pao et al. (Su et al., 2002; Pao et al, 2007), based on three-dimensional elasticity/ piezoelectricity (Ding & Chen, 2001), state-space formalism (Stroh, 1962) and plane wave expansion for the analysis of free waves in multi-layered anisotropic structures. The new formulation of MRRM deals with the exponentially growing and decaying terms separately and refrains from matrix inversion. It is a promising analytical matrix method, which bearing unconditionally numerical stability, for accurately modeling the multi-layered acoustic wave devices (Guo and Chen, 2008b). Numerical methods, including the finite difference method (FDM), the finite element method (FEM), the boundary element method (BEM) and the hybrid method of BEM/FEM (Makkonen, 2005), are powerful for modeling multi-layered acoustic wave devices with complex geometries and boundaries. However, they are less accurate and efficient, especially for high frequency analysis. The reason is that the wave media should be modeled by tremendous elements of small size to ensure computational convergence. Analytical-numerical mixed methods, such as the finite element method/boundary integral formulation (FEM/BIF) (Ballandras et al., 2004) and the finite element method/spectral domain analysis (FEM/SDA) (Hashimoto et al., 2009; Naumenko, 2010), are usually powerful for modeling both the small-sized accessories and the large-dimensioned wave media with high accuracy. They seem to be promising as long as the uniformity of their formulation is improved (Hashimoto et al., 2009; Naumenko, 2010). Although some of these matrix methods are extendable to modeling the multi-layered acoustic wave devices with Bragg Cell, there are few investigations focused on this subject. Few studies have been reported on the effects of a Bragg Cell on wave propagation characteristics of multi-layered acoustic wave devices either. To the authors’ knowledge, all existing references aimed at Bragg Cell in solidly mounted resonators. Zhang et al. (2006, 2008) and Marechal et al. (2008) studied both the resonant transmission in Bragg Cell and acoustic wave propagation in multi-layered bulk acoustic devices with Bragg Cell. Tajic et al., (2010) presented FEM combined with BEM and/or PML to simulate the solidly mounted BAW resonators with Bragg Cell. The formation mechanisms of the frequency bands in Bragg Cell are still an untouched topic. It should be pointed out that the Bragg Cell, as a kind of reliable wave guiding and isolating structure, is potential for utilizing in multi-layered acoustic wave devices working with various acoustic modes including Rayleigh modes, Love modes, Lamb modes, SH modes and bulk longitudinal/transversal modes, so as to improve their performances (Yoon & Park, 2000; Chung et al., 2008). Moreover, for acoustic wave devices working with a specific acoustic mode, other spurious modes inevitably exist. Therefore, for appropriately designing the multi-layered acoustic wave devices with Bragg Cell, modeling methods should be established by considering various wave modes and based on an integrated model, which reckoning on the propagation media, electrodes, Bragg Cell, support layer and substrate. In addition, for appropriately designing the Bragg Cell to improve the performance of multi-layered acoustic wave devices, the features and the mechanisms of frequency bands in the Bragg Cell should be studied. The influence of inserted Bragg Cell on acoustic wave propagation in the working layer should also be clearly revealed.

In this chapter, the wave behavior in the Bragg Cell and the design rules of a Bragg Cell are studied by taking SH wave mode as illustration and by using the Method of Reverberation-Ray Matrix (MRRM). The MRRM is also proposed for accurate analysis and design of multi-layered acoustic wave devices with Bragg Cell, based on an integrated model involving the effects of electrodes, Bragg Cell, support layer and substrate on the working media. Firstly, the MRRM is extended to the analysis of SH wave dispersion characteristics of a ternary Bragg Cell, whose unit cell consisting of three isotropic layers. Based on the resultant closed-form dispersion equations, the formation mechanisms of the SH wave frequency bands are revealed. The design rules of the Bragg Cell according to specific isolation requirements of SH waves are summarized. Secondly, the integrated model, which incorporates the effects of electrodes, Bragg Cell, support layer and substrate on the working piezoelectric media by modeling them as individual non-piezoelectric or piezoelectric layers, is proposed for accurately analyzing acoustic wave propagation in multilayered acoustic wave devices. The formulation of MRRM for the integrated multi-layered structures based on the state space formalism is derived, by which the propagation characteristics of waves can be investigated. In view of the achieved dispersion characteristics, the operating status of various acoustic wave devices can be decided. Thirdly, numerical examples are given to validate the proposed MRRM, to show the features and the formation of SH-wave bands in the Bragg Cell and to indicate the resonant characteristics of multi-layered acoustic wave devices. Finally, conclusions are drawn concerning the SH wave behavior in the Bragg Cell, the advantages of the integrated model and MRRM, and the resonant characteristics of multi-layered acoustic wave devices.

## 2. The features and formation of SH-wave bands in the Bragg Cell

Consider an infinite periodic layered structure with each unit cell containing three isotropic elastic layers. A unit cell is depicted in Fig. 1, which can completely determine the band features of the infinite periodic layered structure by invoking the Floquet-Bloch principle (Mead, 1996). The surfaces and interfaces of the unit cell are denoted by numerals 1 to 4 from top to bottom, and the layers are represented by numerals 1 to 3 from top to bottom. Due to the isotropy of the layers, the in-plane wave motion is decoupled from the out-of-plane one. We limit our discussion to the out-of-plane (transverse) wave motion, i.e. only the SH type mode is present.

### 2.1. SH wave dispersion characteristics of the Bragg Cell

Within the framework of the method of reverberation-ray matrix (MRRM) (Su et al., 2002; Pao et al., 2007; Guo, 2008), constituent layers of the unit cell are individually described in the corresponding local dual coordinates. Fig. 2 depicts the local dual coordinates of a typical layer *JK* or *KJ* will be attached to physical variables of the typical layer

According to the elastodynamics of linear isotropic media (Eringen & Suhubi, 1975), the plane wave solutions to the out-of-plane displacement *JK* (or *j*) in its pertaining coordinates *JK* omitted)

where *J*, with

First, we consider the spectral equations within the layers. The transformed displacement

Substituting Eq. (1) into Eq. (3) and Eq. (2) into Eq. (4), and noticing the functions *j* (*JK* or *KJ*)

where *j* (*JK* or *KJ*), respectively. It is noted that the thickness wavenumber

Second, we consider the spectral equations at the interfaces between adjacent layers. The compatibility of displacements and equilibrium of stresses at the interfaces 2 and 3 are expressed as

Substituting Eqs. (1) and (2) into Eq. (6), one obtains the scattering relations at interfaces

(7) |

Third, we consider the spectral equations at the top and bottom surfaces. The Floquet-Bloch principle of periodic structures (Brillouin, 1953; Mead, 1996) requires that the displacement (stress) of bottom layer at the bottom surface 4 should relate to that of top layer at the top surface

where

Finally, introducing the phase relations of all layers as given in Eq. (5) to the scattering relations of all interfaces and surfaces as given in Eqs. (7) and (9), we obtain the system equations with all the departing wave amplitudes as basic unknown quantities

(10) |

where

The dispersion equation governing the characteristic SH waves in periodic ternary layered media is obtained by vanishing of the determinant of system matrix

.Further expansion of the determinant in Eq. (11) gives the closed-form dispersion relation of characteristic SH waves in periodic ternary layered media as follows

(12) |

Dispersion equation (12) assures unconditionally numerical stability because

(13) |

Define

as the characteristic impedance of SH wave in layer

(15) |

where

signifies the contrast of characteristic impedances of SH waves in layer

### 2.2. Formation mechanisms of SH-wave bands in the Bragg Cell

Based on Eq. (15), in which

Owing to

(18) |

where *j*, but may be imaginary number below the cutoff frequency

where

If

(21) |

which is the dispersion relation for periodic binary layered media already obtained (Shen & Cao, 2000; Wang et al., 2004).

In any circumstances, as

where

where

(24) |

which indicates that the band structures of the periodic ternary layered media are not only determined by the fundamental dispersion curve of the equivalent SH wave according to zone folding effect, but also influenced by three disturbance terms with disturbing functions

(25) |

for coincident and separating points, respectively.

In physics, at any interface

However, the near constructive and destructive interferences specified by Eq. (23) and Eq. (24) can occur in general periodic ternary layered media.

In summary, the occurrence of some specified combination of exact or near constructive and destructive interference phenomena in the unit cell makes the equivalent SH wave travel through the unit cell and gives birth to the pass-bands, whereas the occurrence of other specified combination of exact or near destructive interference makes the equivalent SH wave unable to pass through the unit cell and brings about the stop-bands. Although the above discussion on the formation mechanisms of SH-wave bands is based on the periodic ternary layered structure, it is actually extendable to SH wave in general periodic layered media.

### 2.3. Design rules to the Bragg Cell concerning with SH wave bands

The discussion of formation mechanisms of SH wave bands in the layered Bragg Cell indicates that the contrasts of characteristic impedances of the constituent layers, the characteristic time of the unit cell and the characteristic times of the constituent layers are three kinds of essential parameters, which influence the band properties. First, the contrasts of characteristic impedances decide whether the stop-bands other than that due to SH wave cutoff property exist or not. When the characteristic impedances of all the constituent layers are identical, any SH waves above the maximum cutoff frequency can propagate in the periodic layer without attenuation and no stop-bands other than that due to the SH wave cutoff property exist. In other cases, stop-bands exist above the maximum cutoff frequency, and the contrasts of characteristic impedances decide the widths of the frequency bands. The characteristic time of the unit cell decides the slopes of the dispersion curves of equivalent SH waves, thus it definitely specifies the number of pass-bands/stop-bands in a given frequency range. The characteristic times of the constituent layers mainly decide the mid-frequencies of the frequency bands. It should be pointed out that the mass densities and shear moduli of constituent layers affect all the three kinds of essential parameters, while the thicknesses of the constituent layers only influence the characteristic times of the unit cell and of the constituent layers. These rules can be used for the design of layered Bragg Cells according to the SH-wave bands requirements.

### 2.4. Numerical examples

In this section, the above proposed MRRM for dispersion characteristic analysis and the mechanisms for band structure formation of SH waves in periodic ternary layered media are validated by considering a periodic ternary layered structure with the unit cell consisting of one Pb layer in the middle and two epoxy layers at the up and down sides. The thickness of the Pb layer is 10mm and that of the epoxy layers is 5mm. The material parameters of Pb and epoxy including the Young’s modulus, shear modulus and mass density are ^{3} and ^{3}. The band structures of SH waves in this periodic ternary layered medium are calculated by the formulation presented in Section 2.1. For the convenience of presentation, the results are represented by the dimensionless wavenumbers

We first consider the property of SH-wave band structures in the exemplified periodic ternary layered medium. Figure 3 gives the band structures of SH waves below 140 kHz represented as various forms of graphs. Figure 3(a) depicts the phase constant surfaces in the pass-bands as the dimensionless wavenumbers

It is seen from Fig. 3(a) that the phase constant surfaces in the pass-bands are symmetrical with respect to the vertical plane

In Fig. 3(d), the comparison between the phase constant spectra obtained by our proposed method and those calculated by Wang et al. (2004) indicates good agreement. In Fig. 3(e), the comparison between the

Let us now consider the formation of SH-wave band structures in the exemplified periodic ternary layered structure. We plot in Figs. (4a) and
(4b) the phase and the attenuation constant spectra of characteristic SH waves together with the dispersion curves of the equivalent SH waves in the exemplified periodic ternary layered structure, as

It is clearly seen from Figs. (4a) and (4b) that above the maximum cutoff frequency

It should be emphasized that although the above example is a simple periodic ternary layered structure, the obtained property and formation of SH wave band structures are in fact also applicable to SH waves in all periodic layered isotropic media.

## 3. Analysis of acoustic waves in integrated multi-layered structures

Various multi-layered acoustic wave devices with Bragg Cell can be modeled by the multi-layered structures of infinite lateral extent depicted in Fig. 5, which including both non-piezoelectric layers and piezoelectric layers. Usually, the electrodes, support layers and substrate consist of elastic (non-piezoelectric) layers. The propagation media consist of

piezoelectric single-layer or multi-layers. The Bragg Cell can currently be made of alternate elastic layers such as W and SiO_{2} or alternate elastic and piezoelectric layers such as SiO_{2} and AlN (Lakin, 2005), and may in the future be made of alternate piezoelectric layers. Assume in the multi-layered model, each one of the

### 3.1. Modeling of the non-piezoelectric layers (electrode, Bragg Cell, support layer and substrate)

Based on the three-dimensional linear elasticity (Stroh, 1962), the equations governing the dynamic state of a homogeneous, arbitrarily anisotropic elastic medium in absence of body forces can be written as

where the comma in the subscripts and the dot above the variables imply spatial and time derivatives,

In the case of layer configuration, the state space formalism (Tarn, 2002a) can be adopted to describe mathematically the dynamic state of the medium. Referring to the global coordinate system

### 3.2. Modeling of the piezoelectric layers (propagation media and Bragg Cell)

According to the three-dimensional linear theory of piezoelectricity (Ding & Chen, 2001), the dynamic governing equations for the arbitrarily anisotropic piezoelectric medium in absence of both body forces and free charges are

where

Similar to the arbitrarily anisotropic elastic layer, an arbitrarily anisotropic piezoelectric layer can also be described mathematically by the state space formalism (Tarn, 2002b). In view of the global coordinate system

### 3.3. The state equations and solutions of non-piezoelectric and piezoelectric layers

By virtue of the triple Fourier transform pairs as follows

(31) |

the dynamic governing equations (27) and (28) in the time-space domain can be transformed into those in the frequency-wavenumber domain. The quantity

It is noted that the transformed state vector

where

for a layer of arbitrarily anisotropic elastic material, and

for a layer of arbitrarily anisotropic piezoelectric material, with

According to the theory of ordinary differential equation (Coddington & Levinson, 1955), the solution to the state equation (30) can be expressed, in a form of traveling waves, as

(36) |

where

### 3.4. Reverberation-ray matrix analysis of integrated multi-layered structures

#### 3.4.1. Description of the structural system

Within the framework of MRRM, the physical variables associated with any surface/interface

#### 3.4.2. Traveling wave solutions to the state variables

It is seen from Fig. 6 that the local dual coordinates are both right-handed, thus the state equations for an arbitrary layer

(37) |

#### 3.4.3. Scattering relations from coupling conditions on surfaces and at interfaces

Consider the compatibility of generalized displacements and the equilibrium of generalized stresses on surfaces and at interfaces. The spectral coupling equations on the top surface

where

It should be noticed that halves of all the components in vectors

where

The local scattering relations of top surface, interfaces and bottom surface are grouped together from up to down to give the global scattering relation

where the global arriving and departing wave vectors

(46) |

the corresponding coefficient matrices

It is noticed that the exponential functions in the solutions to the state variables of layers as shown in Eqs. (35) and (36) disappear in the scattering relations, since the thickness coordinates on the surfaces and at the interfaces are always zero in the corresponding local coordinates. This is the main advantage of introducing the local dual coordinates.

#### 3.4.4. Phase relations from compatibility conditions of layers

Considering the compatibility between generalized displacements (generalized stresses) represented in local coordinates *KJ*), we have

where *JK* (*KJ*). Substitution of Eqs. (35) and (36) into Eq. (46), one obtains, by noticing *KJ*)

where

Grouping together the local phase relations for all layers from up to down, one obtains the global phase relation

where

#### 3.4.5. System equation and dispersion equation

The global scattering relation in Eq. (43) and the global phase relation in Eq. (48) both contain

where

If there is no excitation (

which may be solved numerically by a proper root searching technique (Guo, 2008). Thus, the complete propagation characteristics of various waves can be obtained. In particular, the resonant frequency of the multi-layered structures can be obtained as

It should be noted that the above proposed formulation of MRRM (Guo & Chen, 2008a, 2008b; Guo, 2008; Guo et al., 2009) excludes any exponentially growing function and matrix inversion, therefore possesses unconditionally numerical stability and enables inclusion of surface and interface wave modes.

### 3.5. Numerical examples

In this section, the above proposed formulation of MRRM for analyzing the propagation characteristics of various waves in the integrated acoustic wave devices are validated by a bulk acoustic resonator (BAR) consisting of _{2} and _{2} layer as the support medium and

The resonant frequencies, represented by the engineering frequency

From Table 2, it is seen that all component layers in the multi-layered bulk acoustic wave device have obvious influence on the wave propagation characteristics, which validates the necessity to model the multi-layered acoustic wave devices by an integrated model with all components considered. The electrodes generally raise the resonant frequencies in the propagation medium except for the first mode. Adding unit cells of the Bragg Cell and the appending of substrate in the multilayered BAR will reduce the resonant frequencies and increase the number of wave modes in a given frequency range. These findings about the effects of electrodes, Bragg Cell and substrate on wave characteristics in the multilayered acoustic wave devices can be used in the design of these devices.

## 4. Conclusion

The accurate analysis and design of layered Bragg Cell and of multi-layered acoustic wave devices with Bragg Cell are studied by the method of reverberation-ray matrix in this chapter. We obtain the analysis formulation, the features and the formation of SH-wave band structures in layered Bragg Cell and the design rules of layered Bragg Cell according to SH-wave band requirements. A unified formulation of MRRM is attained for the analysis of multi-layered acoustic wave devices modeled by integrated multi-layers consisting of working media, electrodes, Bragg Cell, support layer and substrate. The effects of other components on the resonant characteristics in the working media are gained. All findings are validated by numerical examples. The study in this chapter leads to the following conclusions:

(1) In the SH-wave band structures of layered Bragg Cell, the phase constant spectra in pass-bands and the attenuation constant spectra in stop-bands occur alternately. The phase constant spectra of characteristic SH waves are formed from the dispersion curves of equivalent SH waves due to the zone folding effect and wave interference phenomenon. All the attenuation constant loops as

(2) The proposed MRRM for integrated multi-layered acoustic wave devices is analytical based on distributed-parameter model, yields unified formulation, includes all wave modes and possesses unconditionally numerical stability. It therefore leads to high accurate results at small computational cost and is applicable to complex multilayered acoustic wave devices while combined with a uniform computer program.

(3) The integrated model considers nearly all the components in practical multi-layered acoustic wave devices, which definitely renders accurate wave propagation characteristics for guiding the proper design of and suppresses unfavorable spurious modes in the devices. Generally, the electrodes raise the resonant frequencies, while the Bragg Cell and the substrate reduce the resonant frequencies.

In summary, the MRRM, the understanding of SH wave bands in the Bragg Cell and the integrated modeling of multi-layered acoustic wave devices with Bragg Cell in this chapter will push forward the design of high-performed acoustic wave devices.

### Acknowledgement

This study was financially supported by the National Natural Science Foundation of China (Nos. 10902045 and 11090333), the Postdoctoral Science Foundation of China (Nos. 20090460155 and 201104019) and the Fundamental Research Funds for the Central Universities of China (Grant No. lzujbky-2011-9).