# Angular spectrum method for acoustic beam in anisotropic media and structures

Wei-Jiang Xu, Jinying Zhang, Xianmei Wu, Samuel Dupont

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## To cite this version:

Wei-Jiang Xu, Jinying Zhang, Xianmei Wu, Samuel Dupont. Angular spectrum method for acoustic beam in anisotropic media and structures. 2023 International Congress on Ultrasonics, ICU, Sep 2023, Beijing, China. hal-04470200

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Submitted on 23 May 2025

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PAPER • OPEN ACCESS

# Angular spectrum method for acoustic beam in anisotropic media and structures

To cite this article: Wei-Jiang Xu et al 2024 J. Phys.: Conf. Ser. 2822 012156

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# Angular spectrum method for acoustic beam in anisotropic media and structures

Wei-jiang Xu<fnref n="1" />,<fnref n="2" />, Jin-ying Zhang<fnref n="3" />, Xian-mei Wu<fnref n="4" />,<fnref n="5" />, Qian Cheng<fnref n="6" />, Samul Dupont<fnref n="1" />

Email: wei-jiang.xu@uphf.fr

**Abstract.** The Angular Spectrum (AS) method is commonly used to calculate the propagation of scalar wave fields. This study extends the application of AS to vector wave propagation, specifically addressing acoustic waves in anisotropic media. The approach involves using plane wave spectra to express all possible mode waves, namely one q-P wave and two q-S waves, that propagate in various directions in the medium. These AS are determined by applying boundary conditions at the source surface where the vector waves are generated and the modes waves are coupled. AS decomposition of waves requires the determination of complex wavenumber vectors (or slownesses) through the Christoffel equation, including both propagating and evanescent waves. The determination of the acoustic beam profile in TeO2 crystals is presented as an application example of AS.

## 1. Introduction

Modelling elastic waves for a particular problem involves finding solutions that satisfy both the boundary conditions and the elasto-dynamics equation. For isotropic media, the general solution of the hyperbolic equation can be derived in integral form via the Lamé expression of scalar and vector potentials [1] in its displacement field. However, the Lamé solution is not applicable to anisotropic media. In many physical applications, such as planar ultrasonic sensors, surface wave devices, layered media, and interface reflection/refraction, among others, waves are generated by a planar source situated on a solid surface. In these scenarios, the problem is reduced to a half-space one [2], and the angular spectrum of unbounded plane waves should be a very efficient way to solve the wave propagation in anisotropic materials [3,4].

The solution is considered as the superposition of three orthogonal plane waves (one quasi-compressional wave $qP$ and two quasi Shear waves $qS_1, qS_2$ propagating in all the direction (3 dimensions), while the boundary conditions are satisfied by adjusting the complex amplitude of the plane waves. The reconstruction of an acoustic beam consists of determining a complete set of angular spectrum in the half space by solving the eigen-modes of the Christoffel equation, which

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<footnote n="1">¹ Université Polytechnique Hauts-de-France, CNRS, Université Lille, UMR 8520 IEMN, F-59313 Valenciennes, France</footnote>

<footnote n="2">² INSA Hauts-de-France, F-59313 Valenciennes, France</footnote>

<footnote n="3">³ Beijing Key Laboratory for Precision Optoelectronic Measurement Instrument and Technology School of Optics and Photonics, Beijing Institute of Technology, Beijing, 100811, China</footnote>

<footnote n="4">⁴ State Key Laboratory of Acoustics, Chinese Academy of Sciences, Beijing, 100190, China</footnote>

<footnote n="5">⁵ University of Chinese Academy of Sciences, Beijing, 100049, China</footnote>

<footnote n="6">⁶ Institute of Acoustics, School of Physics Science and Engineering, Tongji University, Shanghai, 200092, China</footnote>
includes both propagating (real wavenumber) and evanescent (complex wavenumber) waves. Particularities related to material anisotropy, such as cuspidal singularity, negative phase velocity, modes coupling at the boundary, can be taken into account.

## 2. Method

<figure><img src="image_9.png" /><figcaption>**Figure 1.** Determination of the wave field in a general anisotropic medium with a plane source using the angular spectrum</figcaption></figure>

Consider an acoustic source ($S_0$) situated on a plane surface (S) (assumed to be the $x,y$ plane for convenience) of an anisotropic substrate with a driven stress or pressure $P_z(x,y)$ (Figure 1.a). In the general case of an anisotropic substrate, three types of waves ($M = \{qP, qS_1, qS_2\}$) corresponding to one quasi-longitudinal and two quasi-transversal modes can be generated. These modes are determined by the Christoffel eigenvalue equation:

$$p_i^M = \frac{c_{ijkl}}{\rho} S_j^M S_k^M p_l^M \quad \text{for } i,j,k,l = \{x,y,z\} \qquad (1)$$

where $p_i^M$ represents the plane wave polarization, and $S_j^M$ is the slowness related to the wavenumber $k_j^M$ by $S_j^M = k_j^M/\omega$. Each type of wave is expressed as a superposition of plane waves of the same type M propagating in all possible direction $\{k_x, k_y, k_z^M\}$ in the substrate (Figure 1.b). The total displacement component of the wave field is given by:

$$u_i(x, y, z, t) = \sum_M \iint_{-\infty}^{\infty} p_i^M U_M(k_x, k_y) \exp[i(k_x x + k_y y + k_z^M z - \omega t)] dk_x dk_y \quad (2)$$

where $U_M(k_x, k_y)$ represents the displacement amplitude of the angular spectrum of type M.

This implies that, under a plane source excitation at the surface, three waves of different natures (modes) can be excited in the anisotropic substrate, each composed of a plane wave angular spectrum. These wave spectra propagate independently without interaction, coupling only at the excitation source and are determined by the source boundary and initial conditions.

### 2.1. Complex slowness or wavenumber determination in substrate half space

In order to construct the angular spectra $U_M(k_x, k_y)$ decomposed at the source surface (x-y plane), it is necessary to determine the wavenumber component $k_z^M$ of the plane wave as a function of those of $k_x$ and $k_y$. Instead of using the Christoffel equation to obtain real wave velocities (or slownesses) in a given direction, equation (1) is employed to solve $k_z^M$ (or $S_z^M$) by substituting two given components of $k_x$ and $k_y$ (or $S_x$ and $S_y$). A sixth-order equation of $k_z^M$ can be obtained, from which three pairs of $k_z^M$ and their associated wave polarization $p_i^M$ can be resolved. The $k_z^M$ values can be real or complex, corresponding to propagating and evanescent waves, respectively. For waves in the anisotropic substrate, only three $k_z^M$ with positive group velocity are chosen for the construction of the angular spectra.
## 2.2. Boundary conditions for determining the angular spectra

After determining the wavenumber component $k_z^M$ and by omitting $\exp(-i\omega t)$ in the harmonic wave case, Eq. (2) can be expressed as:

$$u_i(x, y, z) = \iint_{-\infty}^{\infty} \sum_M p_i^M U_M(k_x, k_y) \exp[i k_z^M(k_x, k_y)z] \exp[i(k_x x + k_y y)] dk_x dk_y \quad (3)$$

which is the Fourier transform of the function $\sum_M p_i^M U_M(k_x, k_y) \exp[i k_z^M(k_x, k_y)z]$

Using the general Hooke's law:

$$\sigma_{ij}(x, y, z) = \sum_{k,l} c_{ijkl} \frac{\partial u_k(x,y,z)}{\partial x_l}, \qquad x_l = \{x, y, z\} \qquad (4)$$

the normal stress $\sigma_{zz}(x, y, z)$ and tangential stress $\sigma_{xy} = (x, y, z)$ can be derived. In the case of a finite size piston driving source, three boundary conditions at the surface (S), $z=0$, are applied:

$\sigma_{zz}(x, y, 0) = P_z$ for $(x, y) \in S_0$, $\sigma_{zz}(x, y, 0) = 0$ otherwise

$\sigma_{xy}(x, y, 0) = 0$ everywhere in S

$n \cdot \nabla u_i(x, y, 0) = 0$ everywhere in S, where n is the surface normal and $\nabla$ gradient

From these conditions, the amplitude of the three angular spectra $U_M(k_x, k_y)$ for $M = \{qP, qS_1, qS_2\}$ is determined, and the acoustic beam or field propagation in the substrate can be reconstructed using Eq. (3). To perform the calculation of $U_M(k_x, k_y)$, a discretization should be applied both at the boundary $(x_p, y_q)$ and in the wavenumber $(k_x^p, k_y^q)$ to get in total 3xPxQ linear equations that has the same number of $U_M(k_x, k_y)$ amplitude to be solved numerically.

# 3. Application to the case of TeO₂ crystal

Tellurium dioxide (TeO₂) is a highly anisotropic crystal extensively utilized in acousto-optics for various applications, including Bragg cell modulators, self-collimating devices [5], multi-wave or collinear interactions [6,7], among others. The precise understanding of the acoustic beam structure, field intensity distribution, propagation direction and possible mode coupling at the boundary is crucial for these applications, given the material's anisotropic property.

An illustrative example is given below, where the acoustics beam profile in a TeO₂ crystal cell excited by a top surface transducer is demonstrated using the angular spectrum approach proposed above.

## 3.1. Elastic and anisotropic properties of TeO₂

The crystal structure of TeO₂ belongs to the 422 symmetry class of the tetragonal system, exhibiting a slight piezoelectricity. The elastic constants and density parameters employed in the following calculation are presented in Table 1, with the piezoelectric property being neglected

<table><caption>**Table 1.** Elastic property of tellurium dioxide (TeO₂)</caption><thead><tr><th colspan="6">Elastic constants (GN/m²)</th><th rowspan="2">Density (kg/m³)</th></tr><tr><th>c<sub>11</sub>=c<sub>22</sub></th><th>c<sub>12</sub></th><th>c<sub>13</sub>=c<sub>23</sub></th><th>c<sub>33</sub></th><th>c<sub>44</sub>=c<sub>55</sub></th><th>c<sub>66</sub></th></tr></thead><tbody><tr><td>55.75</td><td>51.1</td><td>22.9</td><td>106.6</td><td>26.6</td><td>66.0</td><td>5984</td></tr></tbody></table>

Figure 2 presents the slowness and group velocity of TeO₂ of the modes $M = \{qP, qS_1, qS_2\}$ in (001) plane then an in-plane rotation of 45°, with $qP$ mode in red, $qS_1$ in blue and $qS_2$ in green. The $qS_1$ mode is a pure shear mode, while $qP$ and $qS_2$ modes are coupled.

## 3.2. Complex $k_z^M$ or $S_z^M$ determination for modes with positive group velocity

The construction of the angular spectrum in TeO₂ requires first the complex solution of $k_z^M$, or $S_z^M$ in the $(k_x, k_y)$ or $(S_x, S_y)$ plane. To better understand the evolution of $S_z^M$ and the mode coupling,
<figure><img src="image_10.png" /><figcaption>**Figure 2.** Slowness and group velocity surfaces of TeO₂ in (001) plane: *qp* mode (red), *qS*₁ (blue), *qS*₂ (green)</figcaption></figure>

Figure3 illustrates the variation of complex $S_z^M$ with ($S_x, S_y$ = 0), where the ($S_x, S_y$) is taken in (010) plane (Figure 3.a) and in (110) plane (Figure 3.b), respectively. For a given ($S_x, S_y$), there are always six $S_z^M$ solutions: three with positive real part corresponding to waves having positive phase velocity and three negative real part corresponding to waves having positive phase velocity. The real $S_z^M$ is the propagating wave, while $S_z^M$ with an imaginary part corresponds to the decaying evanescent wave. Both can be included in the angular spectrum calculation, but the latter has few contribution after one or several wavelengths of propagation.

<figure><img src="image_11.png" /><figcaption>**Figure 3.** Complex slowness $S_z^M$ of coupled *qp* and *qS*₂ modes (in red-cyan and blue-green), and *qS*₁ mode (in purple-orange) of TeO₂ crystal with ($S_x, S_y$) taken in (110) plane (a), and in (010) plane (b). Left figures of (a) and (b) are the projection of complex $S_z^M$ on its real axis.</figcaption></figure>
Supposing that the substrate is situated in a half space with of $z > 0$, the angular spectrum of each mode should consist of waves with positive group velocity, not positive $S_z^M$. In Figure 3.a, the $qP$ mode with positive group velocity corresponds to the slowness branch marked with $qP^+$ in cyan, the $qS_2$ mode corresponds to the branch marked with $qS_2^+$ in red, and the $qS_1^+$ mode corresponds to the branch in purple. In figure 3.b, the situation is a bit more complicated: the $qP$ mode of the positive group velocity remains one part of the slowness branch marked with $qP^+$ in cyan, while the $qS_2$ mode consists of two branches: one marked with $qS_2^+$ in red another marked with $qS_2^+$ in blue corresponding to the slowness branch with a negative phase velocity. The $S_z^{qS1+}$ branch in violet for the $qS_1^+$ mode remains unchanged as it's a pure shear mode.

To aid in selecting the slowness branches with positive group velocity, Figure 4.a provides an example where the source surface of the TeO₂ substrate is initially situated in (110) plane (similar to the situation in Figure 3.a), then inclined at an angle of $\theta = 2.1$ degrees around the z-axis. Only the $S_z^M$ branches of the $qP$ and $qS_2$ modes are presented in the figure, as $qS_1$ mode is decoupled from the other modes. It's shown that regardless of the mode, the evolution of the $S_z^M$ branch from real to complex always occurs where its group velocity $V_g$ changes from positive to negative. This results in branch interchange between the $qP$ mode and the $qS_2$ mode, and this interchange is achieved through the evolution of the $S_z^M$ branch in its complex domain. To better observer this phenomenon, Figure 4.b shows the interchange of $qP$ and $qS_2$ branches through the complex domain and their evolution under different $\theta$ angles. Finally, in Figure 4.a, the $S_z^M$ branches of the two coupled $qP$ and $qS_2$ modes having positive group velocity are composed of the branches indicated by the red and blue dashed lines.

<figure><img src="image_12.png" /><figcaption>**Figure 4.** Evolution of complex slowness branches of the coupled modes $M = \{qP, qS_2\}$ and their branch interchange through the complex $S_z^M$ domain.</figcaption></figure>

### 3.3. Field profile reconstruction from the angular spectrum

To illustrate field beam reconstruction using the angular spectrum reconstruction, a simple case of three decoupled modes of ($qP, qS_1, qS_2$) is given here. The surface of the TeO₂ substrate lies in the (110) plane tilted by 6 degrees with respect to the z-axis (refer to Figure 5.a). A 3mm-wide line transducer operating at 63.5 MHz is employed as the driven source located on the substrate surface, with its wide perpendicular to the [110] direction. Figure 5.b displays the beam profile of the diffracted acoustic wave of the $qS_2$ mode in the substrate.
<figure><img src="image_13.png" /><figcaption>**Figure 5.** Acoustic beam in the TeO₂. (a) Real slowness S_z of TeO₂ in (Sₓ, Sᵧ = 0) (b) Reconstructed beam profile of the diffracted wave for the qS₂ mode from the angular spectrum.</figcaption></figure>

**4. Conclusion**

The angular spectrum approach is developed to determine the acoustical wave propagation in a general anisotropic medium. It involves expressing three different wave modes, $M = \{qP, qS_1, qS_2\}$, in their plane wave spectrum, which are coupled at the boundary where the waves are generated. This method requires resolving the complex slowness component $S_z^M$ in variation with ($S_x, S_y$). The behaviour of complex $S_z^M$ related to the mode coupling is discussed. The method has applications in various wave propagation situations. The determination of the acoustic beam profile in a TeO₂ acousto-optic modulator is demonstrated in this study. Further applications include: wave propagation in multi-layered structures of anisotropic materials, piezoelectric transducer modeling and design in the case where the piezo-source is adhered on an anisotropic substrate or a buffer lens, MEMS devices design to determine field transmission in a radiation medium where the finite-element approach is difficult to apply in 3D cases, and acousto-optic devices study to evaluate opto-acoustic interaction in anisotropic modulators.

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[5] V. V. Kludzin, V. S. Kulakov, D. R. Pape, S. V. Kulakov, V. Molokok, "A new self-collimating shear acoustic mode in TeO₂ for large time-bandwidth multichannel Bragg cell applications", *IEEE Ultrasonics Symposium*, **1997**, 749-752.

[6] L. Ilchenko, V. Grimalsky, S. Koshevaya, G. Burlak, "Dynamic regimes of three-wave interaction of ultrasonic beams in TeO₂", *Physics Letters A* 303, **2002**, 169-178.

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