Numerical study of acoustic focusing using a bianisotropic acoustic lens Open Access

Recent research on dynamic acoustic material homogenization has shown that the conventional material properties of effective modulus and mass density are not always sufficient for describing materials that have sub-wavelength asymmetry and long-range order.^1–6 For heterogeneous fluids, the homogenized material response can be properly approximated with a constitutive form that couples the acoustic pressure to the particle velocity and the momentum density to the volume strain by means of a coupling parameter known as the Willis coupling vector.^7,8 This form of coupling was first described in elastodynamics by Willis⁹ and has recently been shown to be analogous to bianisotropy in electromagnetic materials,⁷ and thus is often referred to as Willis coupling or acoustic bianisotropy. The presence of Willis coupling and recent theoretical and experimental work exploring how to generate this behavior, as well as associated bounds of physically admissible coupling parameter ranges,^10,11 extend the space of available parameters for the design of materials and surfaces to control acoustic waves. Examples include the design of surfaces to generate extreme anomalous transmission angles¹² and asymmetric scattering patterns from geometrically symmetric scatterers.¹³ 

In this Letter, we leverage the pressure-momentum coupling intrinsic to a bianisotropic fluid medium to design an acoustic lens composed of a linear array of geometrically identical finite-length waveguides filled with the same bianisotropic fluid medium. This is accomplished by varying the orientation of the Willis coupling vector of the fluid in each waveguide to manipulate the phase of a transmitted wave with all other effective material properties held constant. Notably, the lack of analytical models describing guided acoustic wave propagation in a bianisotropic fluid necessitates a numerical approach, which we achieve here using a finite element (FE) model previously derived by the authors,¹³ and using the following design procedure. First, we employ the FE model to calculate the transmitted magnitude and phase of acoustic waves propagating through individual bianisotropic acoustic waveguides for all possible angles of the Willis coupling vector relative to the incident wave direction and geometric parameters. We then use the phase of the wave transmitted through the waveguide to design a linear array of waveguides that functions as a focusing lens. Finally, we solve and validate a full simulation of the entire array as a numerical demonstration of the focusing behavior that one can achieve using a single bianisotropic fluid. This work acts as a proof of concept to demonstrate the ability to use Willis coupling to selectively control acoustic wavefronts and may lead to the future design, optimization, and experimental realization of bianisotropy-based acoustic devices.

The constitutive equations for a bianisotropic fluid are

where p is acoustic pressure, κ is the effective bulk modulus, ε is the volume strain, $ψ→$ is the effective Willis coupling vector, $v→$ is the particle velocity, $μ→$ is the momentum density, and $[ρ]$ is the dynamic effective mass density tensor. In this work, we assume an isotropic mass density tensor such that $[ρ]=ρWI$⁠, where $ρW$ is the scalar mass density of the Willis medium and I is the second-order identity tensor. In the frequency domain, the non-dimensionalized Willis coupling parameter, $W→$⁠, is defined as

where ω is angular frequency and $cW=κ/ρW$ is the speed of sound in the low-frequency limit. An anisotropic wave equation for the acoustic pressure p can be derived by inverting the constitutive equations given by Eq. (1)–(2) in terms of strain and velocity and utilizing the definition of the volume strain rate $ε̇=∇·v→$⁠, and is written in the frequency domain as¹³ 

where $W2=W→·W→$ and k_w = ω/c_w. In order to design a focusing lens using a bianisotropic medium, we utilize a finite element model to discretize and solve Eq. (4). The model uses the weak forms derived by Muhlestein et al.¹³ and is implemented in a finite-element analysis software COMSOL Multiphysics. In particular, we make use of the Weak Form PDE Interface coupled to the Pressure Acoustics Frequency Domain module and perfectly matched layers, which enables the efficient simulation of the bianisotropic acoustic lens and is described in more detail below.

We begin design of the focusing lens by characterizing the transmission properties of finite-height waveguides filled with a bianisotropic fluid, using various waveguide dimensions and boundary conditions. The two-dimensional (2D) computational geometry used to calculate the phase of the transmitted field is shown in Fig. 1(a). An incident plane wave with frequency ω₀ and wavelength λ₀ in a background homogeneous fluid with bulk modulus κ₀ and density ρ₀ impinges on a waveguide of length L and height H containing a bianisotropic fluid with coupling vector $W→$⁠,¹³ whose effective bulk modulus and density are assumed to be identical to the background fluid, i.e., $κW=κ0$ and $ρW=ρ0$⁠. The angle between $W→$ and the y axis, which coincides with the direction of propagation of the incident plane wave, is defined as $θW$⁠. The magnitude of the Willis vector is assumed to be $|W→|=1$ and the complex-valued transmission coefficient through the bianisotropic waveguide is denoted T. Perfectly matched layers (PMLs) are used to absorb the acoustic fields reflected and transmitted from the bianisotropic waveguide. Only homogeneous boundary conditions (i.e., rigid and pressure release) are considered, and rigid boundaries are assumed for the background fluid domains and PMLs. The height H of the waveguide is constrained to be smaller than λ₀ to permit excitation of only the lowest order modes.

We are interested in finding combinations of geometric parameters and boundary conditions that yield a large range of available phase imparted by the waveguides as $θW$ is varied, while maintaining high transmission efficiency. That is, we seek to maximize the quantity $(max(∠T)−min(∠T))$⁠, where the minimum and maximum are found over all possible values of $θW$⁠, while keeping $|T|$ close to 1. Later, we will use the transmission properties obtained here to select a set of coupling vector orientations $θW$⁠, waveguide dimensions, and boundary conditions that produce the desired focusing effect, where the dimensions and boundary conditions are identical for all waveguides and only $θW$ is varied.

The variation of T as a function of $θW$ calculated for a Willis waveguide with length $L=0.94λ0$ is shown in Figs. 1(b)–1(e) for several values of the height H, where panels (b) and (c) and panels (d) and (e) contain the results for rigid and pressure release boundary conditions, respectively. We have assumed $c0=κ0/ρ0=1500$ m/s and $f0=10$ kHz, and thus $λ0=0.15$ m. The bianisotropic waveguide with rigid boundaries exhibits an increase in the range of $∠T$ for increasing H for $0.2λ0≤H≤0.8λ0$ as shown in Fig. 1(c); however, $|T|$ decreases as shown in Fig. 1(b), thereby reducing the focusing gain compared to the incident field. In contrast, for the pressure release case, we find an increase in $|T|$ when H is above the cut-on wavelength (⁠$H≥0.5λ0$⁠), paired with an increase in the range of $∠T$⁠, as shown in Figs. 1(d) and 1(e). While the range of available $∠T$ is not as large as the rigid boundary case, we find that the pressure release case presents a range of waveguide height H suitable for our use between approximately $0.8λ0$ and $1.0λ0$⁠. To explore this part of the design space with finer resolution, we conducted a parameter sweep over H for bianisotropic waveguides with $H∈[0.6λ0 1.0λ0]$⁠, using $0.02λ0$ steps (not shown). While we have not performed a formal optimization, we find that the case $H=0.8λ0$ produces a large range in $∠T$ with high $|T|$ over all values of $θW$⁠, and we therefore select it for the focusing acoustic lens design.

The conventional approach to design a focusing lens¹⁴ using an array of discrete line sources can be achieved with the time transformation

where x_n represents the location of the nth line source from the center of the array in the x direction and y₀ represents the focal distance from the source plane along the y axis. In the frequency domain, this time transformation is equivalent to a change in phase, $ϕn$⁠, for the nth discrete line source, given by

where $k0=ω/c0$ is the wavenumber in the background fluid and d is the distance between the sources in the source plane.

The focusing lens design is achieved by approximating the bianisotropic waveguides investigated in Sec. 3 as discrete line sources and determining the waveguide height, length, and $θW$ required to generate the phase shift provided in Eq. (6) to focus the transmitted field. This approximation provides a means to determine the required parameters, but actual focusing behavior will differ from the idealized point source array design since (i) the pressure in the waveguide will not be uniform and (ii) $|T|$ is not equal for all waveguides of different $θW$ [see Fig. 1(d)]. The smallest admissible focusing distance provided the available range of transmitted phases in Fig. 1(e) for an even number of sources is

In this expression N is the total number of waveguide sources and $Δϕ$ is the range of phase (in radians) available for the waveguides used in the array. The subtraction of $d/2$ accounts for the distance to waveguide center for an even number of sources as is considered in the present case. For convenience, we assume that the focal distance for the lens designed here, y₀, is the minimum possible distance, or $y0=ymin$⁠.

To demonstrate, we design an array of four pressure-release waveguides (N = 4), each of length $L=0.94λ0$ and height $H=0.8λ0$⁠, as previously determined in Sec. 3. We position the waveguides such that the spacing between each waveguide is equal to the height, i.e., d = H. The minimum focal distance $ymin$ from the available bianisotropic waveguide transmission phase determined from the results presented in Fig. 1(e) is found using Eq. (7). Then the phase required for each for the four waveguides forming the lens to focus at a distance $ymin$ is found using Eq. (6). The corresponding Willis vector angles are finally determined by interpolating the data provided in Fig. 1(e), which are $θW1=0°,$ $θW2=39°,$ $θW3=−39°$⁠, and $θW4=0°$⁠. The computational geometry of the lens is shown in Fig. 2(a) and the magnitude of the resulting field is shown in Fig. 2(b).

We find that the focused Willis waveguide array creates a gain of 3.7 dB at the focal point compared to the amplitude of the incident plane wave. The modest gain of the focused field of the case considered here is a result of the small aperture of the lens, which has an aperture of $3.2λ0$ and f-number of 1.4. Though it is not the objective of this work, the focusing gain can be improved by increasing the lens aperture by including more waveguides and increasing the length, L, of each waveguide in order to provide a large range of $∠T$⁠. As a check on the validity of the lens, we compare the computed field to that of four radiating surfaces with uniform pressure across their faces, where the phases are determined from Eq. (6) and are equivalent to the bianisotropic lens. The results of this “ideal” case are in good agreement with the focusing obtained with bianisotropic lens, as shown in Fig. 2(b). This validates the design approach and the use of bianisotropic acoustic fluid waveguides to achieve focusing solely through changes in orientation of the Willis coupling vector.

This work has shown that a 2D acoustic focusing lens can be created using a bianisotropic acoustic fluid in parallel waveguides where the phasing to produce the focused field is produced solely through changes in orientation of the coupling vector $W→$ within each waveguide. The lens was designed using a finite element model of a bianisotropic acoustic fluid. The geometry of the waveguides creating the focusing lens was determined by performing a parametric sweep of the geometric parameters to find a sub-wavelength waveguide height that yields a sufficiently large range of transmitted phase. The entire array was then simulated and found to be in good agreement with an idealized discrete focused array.

To the authors' knowledge, this work is the first to show the use of acoustic bianisotropy to focus an acoustic wavefront and thus demonstrates one potential application of a bianisotropic acoustic response. Future studies are needed to optimize this behavior. For example, focusing gain and focal distances could be improved by obtaining greater phase ranges, which may be achievable through combinations of longer waveguides, stronger Willis coupling, and anisotropic dynamic density. Additional opportunities for future work include experimental realizations of the acoustic lenses described here and detailed theoretical models of bianisotropic acoustic waveguides to aid in their design.

This work was supported by ONR through MURI Grant No. N00014-13-1-0631 and YIP Grant No. N00014-18-1-2335.