# Polarization of Acoustic Waves in Two-Dimensional Phononic Crystals Based on Fused Silica

^{*}

## Abstract

**:**

## 1. Introduction

_{3}), and TAS (Tl

_{3}AsSe

_{3}), as well as various mercury compounds [7,8,9,10]. Nowadays though, a paratellurite (TeO

_{2}) crystal is used as a working material in acousto-optic devices most of the time.

## 2. Theoretical Model and Formulations

_{i}is an elastic wave displacement vector, T

_{ij}is a Cauchy stress tensor characterizing the reaction of the medium to the applied mechanical stress, f

_{i}is the bulk density of external forces, and ρ is the bulk density of the medium in which the elastic wave propagates. The indices take values of i,j = 1,2,3 and obey the Einstein convention.

_{kl}, which is defined as:

_{ij}is proportional to the strain tensor S

_{kl}of the medium:

_{ijkl}is an elastic constants tensor of the medium. The latest equation represents Hooke’s law in the case of an anisotropic medium, in which stresses and strains are related linearly.

_{ijkl}tensor is symmetric with respect to the second pair of indices, in the absence of external forces to the bulk material, the wave equation (Equation (1)) can be written as follows:

_{ijkl}(r) and the bulk density ρ(r) are step functions of coordinates. Since the propagation of elastic waves occurs only in the material of fused silica, these functions take on the values of the corresponding constants of fused silica in the Ω region, and they are equal to zero in the region of cylindrical holes.

_{i}(r) is a periodic function of coordinates, defined in the Ω region of the phononic crystal unit cell, and equal to zero in the region of cylindrical holes; ω = 2πf is the angular frequency of the elastic wave; and k is the wave vector of the elastic wave.

_{x}, k

_{y}, 0).

_{ij}equals:

_{2}-elements as a basis. After choosing the basis functions, the FEM comes down to the optimization problem of minimizing the approximation error of some functional (given by a weak formulation of the original differential equation) on the FEM mesh. An example of the FEM mesh used in our calculcations for a phononic crystal with d/a = 0.6 is shown in Figure 3. According to [17], the described variational problem takes the following form:

_{I}= (2 − δ

_{ij}) • $\tilde{S}$

_{ij}, where δ

_{ij}is the Kronecker symbol.

^{2}. The set of eigenvectors ũ

_{λ}determines the amplitudes and polarization directions of acoustic waves of frequency ω, which can propagate in a phononic crystal. According to Equation (7), the amplitude of the deformation vector is directly affected by the components of the wave vector. Thus, by changing the value of the wave vector within the first zone and solving the integral equation (Equation (8)) for each individual value of the wave vector k, one can obtain the dispersion relation ω(k).

_{IJ}are dimensionless bulk density and elastic constants, respectively, carrying the numerical values of the corresponding physical constants of fused silica, and ρ

_{0}and c

_{0}are quantities that characterize the dimensions of those physical constants. In dimensionless quantities, the integral equation (Equation (8)) takes the form:

## 3. The Method for Calculations of the Phononic Crystal Acoustic Characteristics

**k**) connects the frequencies and wave vectors of acoustic waves that can propagate in a medium. By fixing the frequency at ω

_{0}= 2πf

_{0}, it is possible to obtain sets of wavenumbers k

_{x}(f

_{0}) and k

_{y}(f

_{0}) of all acoustic waves of frequency f

_{0}that can propagate in a phononic crystal. At a given ultrasound frequency f

_{0}, the wavenumbers k

_{x}and k

_{y}are proportional to the components of the inverse phase velocity of acoustic waves, since k

_{x}= 2πf

_{0}/V

_{x}and k

_{y}= 2πf

_{0}/V

_{y}. Thus, the line of the frequency contour f

_{0}of the dispersion dependence defines the cross sections of the acoustic slowness surface S(φ) in the XY plane of the phononic crystal, where φ is the polar angle. In an acousto-optic device, the isofrequency f

_{0}is the frequency of ultrasound excited by the piezoelectric transducer.

_{0}= 50 MHz, while the projection of those contour lines to the first Brillouin zone can be seen in Figure 4b. Similar to the dispersion surfaces, the isofrequency contour lines are symmetric with respect to the first irreducible Brillouin zone Γ-X-M-Γ. For isofrequencies f

_{0}< 50 MHz, the contour line of each individual acoustic mode is a closed curve. For isofrequencies f

_{0}> 50 MHz, where the dispersion surfaces become nonmonotonic, the contour lines become piecewise interval lines. The calculation of acoustic characteristics at these higher isofrequencies requires a separate analysis. This work presents the case in which the isofrequency is chosen to be f

_{0}= 50 MHz.

**ũ**

_{λ}(

**r**) of the integral problem (Equation (8)) define the amplitudes and polarization directions of ω frequency acoustic waves, which can propagate in a phononic crystal. Since eigenvectors

**ũ**

_{λ}(

**r**) are periodic functions of coordinates, the polarization direction of each acoustic mode varies within the unit cell. Therefore, from a practical point of view, it seems necessary to carry out the calculation of the polarization vector

**ũ**

^{0}averaged over the unit cell region. The components of the averaged polarization vector can be obtained as:

**ũ**

^{0}are defined to within an overall sign, as described in Equation (12). It is clear that in the limit d/a → 0, the averaged solution of Equation (8) should converge to that of solid isotropic fused silica. The requirement of the physical behavior in the mentioned limit makes it possible to choose the sign in Equation (12). We also assume that with a small change in the phononic crystal geometry, that is, with a small change in d/a, the change in the polarization vector direction is continuous. With this assumption, the signs of the components of the averaged polarization vector in Equation (12) can be determined for any values of the ratio d/a.

## 4. Results and Discussion

_{11}≡ c

_{1}= 7.85·10

^{10}N/m

^{2}, c

_{12}≡ c

_{6}= 1.61·10

^{10}N/m

^{2}, and ρ = 2203 kg/m

^{3}.

#### 4.1. Inverse Phase Velocity Distributions

#### 4.2. The Distributions of Walk-Off Angles

_{max}, of acoustic modes in phononic crystals with different normalized hole diameters are presented in Table 2, as well as the propagation directions of the modes, φ *, leading to the maximum walk-off angle.

#### 4.3. The Polarizations of Acoustic Modes

**ũ**

^{0}(φ) of acoustic modes, depending on their propagation direction, featuring two different geometries of a phononic crystal. The averaged polarization vector is calculated using the method described in the previous section. The results show that the fast shear mode is polarized along the Z axis of the cylindrical holes of the phononic crystal. The slow shear acoustic mode appears purely transverse for propagation directions $\phi =\pi n/4,\hspace{1em}n\in \mathbb{N}$ in the XY plane. The longitudinal acoustic mode of a phononic crystal is purely longitudinal when propagating along the indicated directions. This property of the averaged polarization vector is associated with the symmetry of the phononic crystal unit cell.

^{0}(φ) between the averaged polarization vector

**ũ**

^{0}(φ) and the direction of the mode φ propagation. Same as the averaged polarization vector, this angle is a function of the acoustic wave propagation direction. By definition of the corresponding terms, for quasi-shear waves, the value of this angle lies in the region 45° ≤ γ

^{0}< 90°; for quasi-longitudinal waves, it lies in the region 0° < γ

^{0}≤ 45°. The polarization angle γ

^{0}of the fast shear isotropic mode of a phononic crystal is 90° for all directions of wave propagation. This mode is purely shear, and it is polarized along the Z axis of the cylindrical holes for all phononic crystal geometries.

^{0}≤ 90° range. The polarization angle of the longitudinal mode takes the values 0° ≤ γ

^{0}≤ 20.1°. For the phononic crystal geometry in which d/a = 0.8, the polarization angle of the slow shear mode lies in the range of 61.6° ≤ γ

^{0}≤ 90°, as seen in Figure 7b. For the longitudinal mode of a phononic crystal, the range is 0° ≤ γ

^{0}≤ 28.4°. Thus, the anisotropic acoustic modes of a phononic crystal are quasi-transverse and quasi-longitudinal. The transformation of the acoustic mode from quasi-transverse to quasi-longitudinal and vice versa is impossible in a phononic crystal. The limitation is associated with the symmetry of the phononic crystal unit cell, as well as with the assumption that the averaged polarization vector

**ũ**

^{0}(φ) is continuous.

**ũ**

_{λ}. However, this symmetry is not the reason for the orthogonality of the polarization vectors of the phononic crystal acoustic modes, since for any given propagation direction φ, the solution

**ũ**

_{λ}is sought at various wavenumbers k

_{x}and k

_{y}. The eigenvalue of the integral equation (Equation (8)) for different acoustic modes matches because these modes are obtained by isocontouring the dispersion surfaces of a fixed isofrequency f

_{0}. However, when searching for the averaged polarization vectors of acoustic modes, different integral equations with different kernels are solved. Thus, in the low-frequency region of excited acoustic modes, up to 50 MHz, a phononic crystal effectively behaves as a homogeneous anisotropic medium in the sense that there are exactly three acoustic modes in it, and their mean polarizations are mutually orthogonal.

## 5. Conclusions

_{⊥}= 3763 m/s of the shear acoustic mode of solid isotropic fused silica. It is demonstrated that under certain conditions, exactly three acoustic modes propagate in a phononic crystal. In this case, the averaged polarization vectors of acoustic modes are mutually orthogonal for any directions of wave propagation. Thus, under certain conditions, a phononic crystal effectively behaves as a homogeneous bulk anisotropic medium.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Acoustic characteristics of a paratellurite crystal: (

**a**) distributions of the inverse phase velocities of acoustic modes in the XY plane and (

**b**) distributions of the walk-off angles of the acoustic modes’ energy in the XY plane.

**Figure 2.**Model of a two-dimensional square-lattice phononic crystal: (

**a**) three-dimensional physical model, (

**b**) unit cell, and (

**c**) unit cell, made dimensionless.

**Figure 3.**An example of the finite element method mesh used in calculation for a phononic crystal with d/a = 0.6.

**Figure 4.**Band structure of a two-dimensional phononic crystal with a normalized hole diameter d/a = 0.8: (

**a**) the first three dispersion surfaces corresponding to the three lowest acoustic modes in a phononic crystal and (

**b**) projection of the dispersion surface contour lines onto the first Brillouin zone at an isofrequency f

_{0}= 50 MHz.

**Figure 5.**(

**a**) Inverse phase velocity curves in the XY plane in an isotropic material of fused silica. Shown in (

**b**–

**e**) are inverse phase velocity curves of a phononic crystal with the normalized hole diameter: (

**b**) d/a = 0.2; (

**c**) d/a = 0.4; (

**d**) d/a = 0.6; and (

**e**) d/a = 0.8.

**Figure 6.**Distribution of acoustic energy walk-off angles in the XY plane of a phononic crystal with a normalized hole diameter: (

**a**) d/a = 0.4; (

**b**) d/a = 0.6; and (

**c**) d/a = 0.8.

**Figure 7.**Distribution of the averaged polarization vector of acoustic modes in a phononic crystal with a normalized hole diameter: (

**a**) d/a = 0.6 and (

**b**) d/a = 0.8.

**Table 1.**Phase velocities of acoustic modes and their anisotropy coefficients for phononic crystals with various normalized hole diameters.

d/a | Slow Shear Wave | Isotropic Wave | Longitudinal Wave | ||
---|---|---|---|---|---|

${\mathit{V}}_{\mathit{m}\mathit{i}\mathit{n}},\mathbf{m}/\mathbf{s}$ | ${\mathit{V}}_{\mathit{m}\mathit{a}\mathit{x}},\mathbf{m}/\mathbf{s}$ | ${\mathit{V}}_{\mathit{i}\mathit{s}\mathit{o}},\mathbf{m}/\mathbf{s}$ | ${\mathit{V}}_{\mathit{m}\mathit{i}\mathit{n}},\mathbf{m}/\mathbf{s}$ | ${\mathit{V}}_{\mathit{m}\mathit{a}\mathit{x}},\mathbf{m}/\mathbf{s}$ | |

0.2 | 3707 | 3708 | 3709 | 5808 | 5816 |

χ ≈ 1 | χ ≈ 1 | ||||

0.4 | 3191 | 3392 | 3542 | 5315 | 5438 |

χ = 1.13 | χ = 1.05 | ||||

0.6 | 2481 | 3174 | 3310 | 4590 | 5007 |

χ = 1.64 | χ = 1.19 | ||||

0.8 | 1594 | 2976 | 3004 | 3759 | 4536 |

χ = 3.49 | χ = 1.46 |

**Table 2.**Maximum values of the energy walk-off angles of acoustic modes in phononic crystals with various normalized hole diameters.

d/a | Slow Shear Mode | Longitudinal Mode | ||
---|---|---|---|---|

ψ _{max}, ° | φ *, ° | χ_{max}, ° | ψ *, ° | |

0.4 | 6.9 | 23.3 | 2.62 | 22.7 |

0.6 | 25.8 | 24.6 | 9.6 | 22.2 |

0.8 | 51.8 | 27.4 | 19.7 | 21.5 |

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**MDPI and ACS Style**

Marunin, M.V.; Polikarpova, N.V.
Polarization of Acoustic Waves in Two-Dimensional Phononic Crystals Based on Fused Silica. *Materials* **2022**, *15*, 8315.
https://doi.org/10.3390/ma15238315

**AMA Style**

Marunin MV, Polikarpova NV.
Polarization of Acoustic Waves in Two-Dimensional Phononic Crystals Based on Fused Silica. *Materials*. 2022; 15(23):8315.
https://doi.org/10.3390/ma15238315

**Chicago/Turabian Style**

Marunin, Mikhail V., and Nataliya V. Polikarpova.
2022. "Polarization of Acoustic Waves in Two-Dimensional Phononic Crystals Based on Fused Silica" *Materials* 15, no. 23: 8315.
https://doi.org/10.3390/ma15238315