#### Abstract

Phononic crystals (PCs) can be used as acoustic frequency selective insulators and filters. In a two-dimensional (2D) PC, cylindrical scatterers with a common axis direction are located periodically in a host medium. In the present paper, the layer multiple-scattering (LMS) computational method for wave propagation through 2D PC slabs is formulated and implemented for general 3D incident-wave directions and polarizations. Extensions are made to slabs with cylindrical scatterers of different types within each layer. As an application, the problem is considered to design such a slab with small sound transmittance within a given frequency band and solid angle region for the direction of the incident plane wave. The design problem, with variable parameters characterizing the scatterer geometry and material, is solved by differential evolution, a global optimization algorithm for efficiently navigating parameter landscapes. The efficacy of the procedure is illustrated by comparison to a direct Monte Carlo method.

#### 1. Introduction

Just as photonic crystals can be used to manipulate light, phononic crystals (PCs) with inclusions in a lattice with single, double, or triple periodicity can be used to manipulate sound [1]. When a sound wave of a certain frequency penetrates the PC, the energy is scattered by the inclusions. According to Braggโs law, constructive and destructive interference appears in certain directions. It follows that transmission and/or reflection for certain frequencies can be absent (band gaps), even for all angles of incidence (absolute band gaps). The band gaps are ideal and fully developed only in space-filling PCs. In a PC slab with finite thickness, the โband-gap wavefieldsโ are reduced significantly but they do not vanish throughout the gap. Acoustic frequency selective insulators and filters are possible applications.

Several computational methods have been adapted and developed to study wave propagation through PCs. Two such methods are the purely numerical finite-difference time domain (FDTD) method and the semianalytical layer multiple-scattering (LMS) method, which is developed from Korringa-Kohn-Rostoker theory [2]. These methods complement each other. Advantages with the LMS method are its computational speed, which makes it useful for forward modeling in connection with extensive optimization computations, and the physical insight it provides. It appears that the LMS method was first developed for the 3D case with spherical scatterers [3, 4], and recent review papers include Sigalas et al. [5] and Sainidou et al. [6].

The LMS method has also been applied to the 2D case with infinite cylindrical scatterers. It is mainly the in-plane propagation case that has been considered [7โ11], for which one space dimension can be eliminated in the wave equations. Out-of-plane propagation has been treated by Mei et al. [12] and, for band structure calculations, somewhat earlier by Wilm et al. [13] (using the plane-wave method) and by Guenneau et al. [14].

In the present paper, basic LMS equations for propagation of plane waves of any direction through a 2D PC slab are first provided in Section 2. The Poisson summation formula and the Graf addition theorem are utilized. As shown in Section 3, different types of scatterers at the same interface can be allowed, which represents an extension as compared to the treatment in [12]. Coupled equation systems are derived for the different scatterer types. For 3D PC slabs, a corresponding extension has recently proved useful for applications to design of anechoic coatings [15].

Applications to design of 2D PC slabs with broad transmission gaps for incident waves of in-plane as well as out-of-plane directions are presented in Section 4. The design problem is equivalent to a nonlinear optimization problem, and differential evolution [16], a global optimization technique from inverse theory, is used. A 2D PC slab example from Mei et al. [12], with lead cylinders in an epoxy host, is revisited. Geometrical and material parameter values are varied to position and widen its band gap.

#### 2. Basic 2D Layer Multiple-Scattering Computational Method

As in Figure 1, a right-hand Cartesian coordinate system is introduced in a fluid-solid medium surrounded by homogeneous half-spaces. The horizontal directions are and . The medium is periodic with period in the direction and uniform in the *y* direction.

Sound waves with time dependence , to be suppressed in the formulas, are considered, where is the angular frequency. It follows that an incident plane wave with horizontal wavenumber vector will give rise to a linear combination of reflected and transmitted plane waves with displacement vectors

Here, , for a wave of type P,SV,SH, respectively, for a wave in the positive (negative) direction, and

where **g** belongs to the reciprocal lattice

with running over the integers. Furthermore, is the compressional-wave velocity and are the shear-wave velocity . The angular variables of are defined by (2), with a possibly complex . The vectors are defined by , ,โโ. It is convenient also to introduce the compressional- and shear-wave wavenumbers and .

As detailed in [17], for example, and references therein, reflection and transmission matrices , and , can now be introduced, for the discrete set of waves specified by (1)โ(3). The mentioned reference concerns the doubly periodic case for a 3D PC, with periodicity in the direction as well, but the R/T matrix formalism is the same. There are three scatterer interfaces in the illustration of Figure 1. Individual matrices can be combined recursively [18, 19]. Layer thicknesses, as well as translations of individual scatterer interfaces in the *x* direction, are conveniently accounted for by phase shifts of the complex amplitudes of the plane-wave components.

##### 2.1. Interface with Periodically Distributed Scatterers of a Common Type

Explicit expressions for the matrices are well known for an interface between two homogeneous half-spaces [19]. To handle an interface with periodically distributed scatterers, as one of the three in Figure 1, the following cylindrical vector solutions to the wave equations can be used [12]:

where cylindrical coordinates are used according to , and . The index , and is a real number. Residual wavenumbers , are defined by

The notation , , and , , is used for the two basic cases with as the Bessel function and as the Hankel function , respectively.

For scatterers at

where is the lattice period and runs over the integers, and for an incident plane wave as in (1), the total scattered field can be written as (cf. [3]) where is the component of .

The vector is determined by solving the equation system

where is the appropriate identity matrix, gives the coefficients for expansion of the incident plane wave in regular cylindrical waves , is the lattice translation matrix , and is the transition matrix for an individual scatterer. Specifically, and where gives the coefficients for expansion in regular cylindrical waves of the scattered field from all scatterers except the one at the origin.

The R/T matrices are obtained, finally, by transforming the expansion (9) to plane waves of the type (1). Specifically, (9) can be rewritten as The sign in is given by the sign of .

Explicit expressions for the coefficients in (11) are readily obtained from (9) and (4)โ(6) by invoking, for and , the relation

where with Im . The relation (12) is valid for , and it can be verified using the Poisson summation formula.

##### 2.2. Computation of the Expansion Coefficients , the Matrices ฮฉ, and the Matrices T

An incident plane compressional wave , where , can be expanded as

Noting that , this follows readily from the well-known Bessel function relation .

An incident plane shear wave of SV or SH type can be expanded by a superposition of two cases. The first case, with and , can be expanded as

The second case, with as before and , has the expansion

The lattice translation matrix can be determined by applying, for and , the relation where the last sum is taken over all integers and This relation follows from the Graf addition theorem [20] for Bessel functions. The elements of the lattice translation matrix vanish unless . The remaining elements depend on and through only. Explicitly, Moroz [21] has published a representation of lattice sums in terms of exponentially convergent series, which has been used in the present work for numerical evaluation of the quantities defined in (17).

For a homogeneous cylindrical scatterer, the interior field and the exterior field can be expanded in cylindrical waves and , respectively. An equation system for the -matrix elements , which depend on , is then readily obtained from the standard boundary conditions concerning continuity of displacement and traction at the cylinder surface. The scatterer as well as the host medium can be either fluid or solid. Because of the circular symmetry with a cylindrical scatterer, scattering only appears to the same component . Details concerning the case , for which the -SV and SH solutions decouple, are given by Mei et al. [9].

#### 3. Different Types of Scatterers at the Same Interface

The LMS method is commonly applied for lattices with identical cylindrical scatterers within the same layer. As shown below, however, different types of scatterers within the same layer can also be accommodated. A similar extension for the restriction to in-plane wave propagation is made in Ivansson [22].

An illustration is given in Figure 2. Centered at the same level, at each of the three scatterer interfaces, there are two types of cylindrical scatterers. Each scatterer interface is treated separately. Choosing coordinates appropriately, scatterers of the first type, with transition matrix and scattered-field expansion coefficients denoted , appear at , for integers . Scatterers of the second type, with transition matrix and scattered-field expansion coefficients denoted , appear at points in between, that is, . The reciprocal lattice vectors become , where *m* runs over the integers.

The generalization of the expression (9) for the scattered field becomes

It follows that where, for a scatterer of the first type at , and give the coefficients for expansion in regular cylindrical waves of the scattered field from all other scatterers of the first and the second types, respectively. The vectors and are defined analogously. For a scatterer of the second type at , and thus give the coefficients for expansion in regular cylindrical waves of the scattered field from all other scatterers of the second and the first types, respectively.

With , it follows that and . For a certain matrix , to be determined, and . The equation system for determination of and becomes

In order to form the R/T matrices, incident plane waves with different horizontal wavenumber vectors have to be considered, where belongs to the set providing the reciprocal lattice. Noting that the union of the scatterer positions is a small square lattice with period , the following expression for as a difference of matrices is directly obtained
Only those in for which is even are reciprocal vectors for the small lattice with period . Since a lattice translation matrix is periodic in its first argument with period , there will be two groups of with different matrices according to (22). Specifically,
pertains to with even , and
pertains to = (2*ฯ **m*/*d*, 0, 0) with odd *m*.

The transformation of the expansion (19) to plane waves of the type (1) can be done separately for each of the and sums. In the latter case, the translation from the origin causes a sign change for some combinations of incident and scattered reciprocal lattice vectors. Specifically, with and , the double sum corresponding to the one in (11) appears as

#### 4. Designing 2D PC Slabs with a Transmission Gap

It is well known that band gaps can appear when scatterers with a large density are arranged periodically in a host with a small density. A particular 2D PC slab example with lead cylinders in epoxy was considered in Mei et al. [12]. The epoxy parameters were 2540 and 1159.817โm/s for the compressional- and shear-wave velocities, respectively, and 1.18โkg/dm^{3} for the density. Corresponding lead material parameters were 2160 and 860.568โm/s, and 11.4โkg/dm^{3}. The slab was formed by sixteen layers of lead cylinders, each with a radius of 3.584โmm, with a spacing between cylinder centers of 11.598โmm in the as well as directions, cf. Figure 1 where there are three layers. A band gap centered at about 60โkHz was found. (Dimensionless frequencies and distances were actually used in the paper, but a specialization is made here.)

Figure 3 is a contour plot of the transmittance, that is, time- (and space-) averaged transmitted energy flux relative to the incident one, up to 120โkHz. The vertical axis is for the angle , where the direction vector of the incident compressional plane wave is (, , ). In-plane and out-of-plane incidence angles are considered together in Figure 3, where the upper half with concerns incidence in the plane (in-plane propagation) and the lower half with concerns incidence in the plane. Only the first Brillouin zone is involved, since when equals the epoxy compressional-wave velocity 2540โm/s, โmm, and the frequency is less than โkHz. Of course, the band gap from Mei et al. [12], at about 60โkHz, shows up clearly in this kind of plot. One might wonder how the geometrical and/or material parameters of the slab should be modified to achieve a prescribed desired change of the appearance of the gap.

Global optimization methods can be used to design PC slabs with desirable properties. Simulated annealing, genetic algorithms and differential evolution (DE) are three kinds of such methods, that have become popular during the last fifteen years. DE, to be applied here, is related to genetic algorithms, but the parameters are not encoded in bit strings, and genetic operators such as crossover and mutation are replaced by algebraic operators [16].

As a very simple example, to try to position and widen the gap in Figure 3, an objective function for DE minimization is specified as the maximum transmittance in the frequency band 45โ65โkHz when the incidence angle is varied between and for the two azimuthal angles (in-plane propagation in the plane) and (propagation in the plane). A configuration with cylinders of two alternating sizes is allowed, as depicted in Figure 2. Still, the slab is formed by sixteen layers of cylinders. The following seven parameters are varied within the indicated search space: scatterer compressional-wave velocity [1500โm/s 3000โm/s], scatterer shear-wave velocity [650โm/s 1200โm/s], scatterer density [7โkg/dm^{3 } 14โkg/dm^{3}], layer thickness [8 16โmm], the largest scatterer radius [0.225โ โค 0.4โ], the smallest scatterer radius [0.45โ], the lattice period [8()/3 4 ()]. The layer thickness is the distance between subsequent scatterer interfaces (cf. Figure 2).

Table 1 shows the optimum obtained with DE, along with corresponding parameter values. A reduction of the transmitted field with more than 160โdB is achieved throughout the band 45โ65โkHz and throughout the solid angle intervals for , for the direction of incidence. High-velocity high-density cylinders seem preferable, since the lead velocities and density are all increased to values close to the upper ends of the corresponding search intervals. In this case, as large as possible values of , , and could in fact have been fixed from the start. Moreover, the cylinders are almost as densely packed horizontally as allowed by the search interval for . Hence, the optimization could in fact have been simplified considerably. With only a few free parameters, a complete search, for example, can be a feasible alternative.

Compared to Figure 3, the contour plot in Figure 4 of the transmittance for the optimized PC slab shows a band gap around 60โkHz that has become significantly wider and also deeper. The allowed increases of , , , and ()/ are certainly essential for producing this effect. Variation of , , and is needed to position the band gap at the desired frequency interval. Although the optimization was performed with the restriction , small transmittance is apparently achieved for all angles ().

It turns out that the transmittance for the optimized PC slab remains small within the 45โ65โkHz band for all out-of-plane incidence angles (an โabsoluteโ band gap). Figure 5 shows the transmittance in the same way as Figure 4, but for changed to and changed to . The upper and lower halves exhibit increased symmetry, since the angles involved are closer to one another.

Apparently, the optimized PC slab has large transmittance below about 30โkHz and there are regions with large transmittance above 80โkHz as well. The band gap with small transmittance extends to higher frequencies, than those in the band 45โ65โkHz, for plane-wave incidence directions with either large or small .

The efficacy of the DE technique can be illustrated by showing the decrease of the maximum transmittance within the specified frequency/angle region as the number of tested parameter settings for the PC slab evolves. Figure 6 shows this decrease for the example from Figure 4. A comparison to the much less efficient MonteCarlo method, with random selection of parameters from the search space, is included. The low efficiency of the MonteCarlo approach shows that the desired PC slabs with small transmittance only appear in a small portion of the search space. For example, only about 0.15% of the random slab selections had a maximum transmittance below โ20โdB within the specified frequency/angle region. About 0.01% had a maximum transmittance below โ100โdB.

The difference between and in Table 1 is rather small. Optimization was also tried with all cylinders of exactly the same radius, as in Figure 1. The obtained optimum, with cylinders of radius 4.2503โmm and slightly modified values for the other parameters, was almost as good as the one presented in Table 1. The advantages with cylinders of different sizes are expected to be more significant in more complicated filtering cases, involving, for example, more than one frequency band.

#### 5. Conclusions

The layer multiple-scattering (LMS) method is a fast semianalytical technique for computing scattering from layers including periodic scatterer lattices. For the 2D case with cylindrical scatterers and any solid angle direction of an incident plane wave, an extension has been made to scatterer lattices with cylindrical scatterers of two different sizes in the same horizontal plane.

Global optimization methods from inverse theory are useful for designing PC slabs with desirable properties. A differential evolution algorithm has been applied here to position and widen an โabsoluteโ band gap for a certain 2D PC slab. Although only limited angle intervals for the direction of incidence were included for the optimization, small transmittance was achieved for all solid angles specifying the direction of the incident wave.

The possibility to include cylindrical scatterers of two different sizes in the same horizontal plane provides an additional degree of freedom that can be useful for PC design purposes. In the presented example, with its specification of objective function and search intervals for the parameters, however, the difference between the optimal cylinder radii was rather small and the improvement only marginal. The additional flexibility is expected to be more important in more complicated filtering applications. For example, specified regions in frequency-angle space with large transmittance could be desired in addition to specified regions with small transmittance.

#### Acknowledgment

Alexander Moroz kindly provided his Fortran routine for calculation of lattice sums.