Broadband ultra-long acoustic jet based on double-foci Luneburg lens

An acoustic jet is an acoustic focal field with a subwavelength full width at half maximum (FWHM) while maintaining a long propagation distance, which has potential applications in structural health monitoring,¹ medical imaging,² and energy harvesting.^3–5 With the development of acoustic metamaterials for the control and manipulation of the propagation of acoustic waves in recent years,^6–8 using acoustic metamaterials to generate the acoustic jet has attracted much attention. For example, Minin et al. theoretically proposed to use an acoustic metamaterial lens to focus the acoustic energy for achieving the acoustic jet.⁹ Maznev et al. proposed to modify the aperture size of an acoustic metamaterial lens to achieve the acoustic jet, which was capable of focusing at least 50% of the incident power.¹⁰ Lopes et al. experimentally demonstrated the use of an acoustic metamaterial lens for achieving an acoustic jet with a FWHM smaller than λ/2 (λ is the wavelength).¹¹ Furthermore, Canle et al. proposed a practical realization of the subwavelength acoustic jet by properly designing the metamaterial lens and choosing the lens and background medium materials.¹² Most of the existing methods for creating acoustic jets have a limitation on the working distance due to the small focal area. More recently, Lu et al. designed a gradient index (GRIN) acoustic metamaterial generalized Luneburg lens (GLL), which can achieve a super long working distance up to 17λ.¹³ However, this GRIN lens can only achieve a super long acoustic jet in the far field or short acoustic jet in the near field.

The Luneburg lens is a spherically symmetric GRIN lens, which can focus incoming plane waves onto the outer surface of the opposite side of the lens.^14–18 The optical double-foci Luneburg lens was initially proposed in 1984 by Sochacki,¹⁹ which was derived from the standard optical Luneburg lens.²⁰ By tailoring the refractive index distribution, a modified generalized Luneburg lens (MGLL) can be designed to achieve two focal points at arbitrary positions, which allows a high power flow between the two focal spots. For example, Mao et al.²¹ explored a tuneable photonic nanojet formed by using the MGLL, which can achieve ultra-long subwavelength focusing between the two focal points. At microwave frequencies, Chou et al.²² investigated the use of a metasurface comprising circular metallic patches printed on a grounded dielectric substrate to achieve the double-foci MGLL. Inspired by the optical double-foci GRIN MGLL,¹⁹ in this study, an acoustic double-foci GRIN MGLL is proposed for achieving a super long acoustic jet that covers both the near field and far field. The variation of the refractive index of the lens is achieved by varying the filling ratio of the lattice unit cells, which allows for tailoring the velocity of the acoustic wave propagation in the structure. With this method, the graded change of the refractive index can be obtained in a broadband frequency range.

The double-foci MGLL design has two concentric circles (radii of R and R′, R is the radius of the lens) with different graded refractive index profiles. When a line source is used to generate plane waves that interact with the lens, the acoustic waves passing through the MGLL will produce double-foci as shown in Fig. 1(a). All of the rays entering the inner circle will be perfectly focused on the near field with a focal length of F₁ (blue dot), whereas the rays entering the outer circle will be focused to the far field with a focal length of F₂ (red dot). According to Ref. 19, the trajectories of rays can be described by the following set of equations:

where r is the radial distance, $k=nrr sin(φ)$⁠, and $φ$ is the angle between the tangential direction and the position vector r. P_a is a parameter between 0 and 1, which can be used to determine the radius of the inner circle of the double-foci Luneburg lens. P_a is related to the refractive index, and the radius of the inner circle $R′$ can be described as $Pa=R′n(R′)$⁠, where $n(R′)$ is the refractive index at the inner circle. $ρ=rn/R$⁠, and then the refractive index is obtained as

where $ω1(ρ,F2)$=$(1/π)∫ρ1arcsin(k/F2)/k2−ρ2dk$ and $ω2ρ,Fi,Pa=(1/π)∫ρPaarcsin(k/Fi)/k2−ρ2dk$⁠.

Based on the distributions of the refractive index in Eq. (2), it can be seen that the refractive index is a function of both the near-field focal length F₁ and far-field focal length F₂. In this study, we fix the value of F₁ and tune the length of the acoustic jet via changing the value of F₂. The refractive index distribution of the MGLL with respect to the radial distance r at different values of F₂ (1.6R, 2.5R, and 3.2R) is provided in Fig. 1(b).

In this study, a perfect impedance-matching double-foci Luneburg lens is designed with continuous variation of the refractive index according to Eq. (2) to generate an acoustic jet. Numerical simulations were performed to investigate the acoustic jet performance. The lens design parameters were chosen to be R = 0.1 m, P_a = 0.5, F₁ = 1.2R, and different values of F₂ (1.6R, 2.5R, and 3.2R) were used. The dimension of the air area in the simulation was 7R × 3R. The frequency domain analysis was performed by using the acoustic module of the COMSOL software (COMSOL Inc., Stockholm, Sweden) at the frequency f = 17 kHz (wavelength $λ=0.02$ m). A line source located at x = −1.5R (y is from −1.5R to 1.5R) was used for the excitation. In the simulations, the radiation boundary conditions were applied on the air boundary to reduce the boundary reflections. The numerical simulation results are shown in Figs. 2(a)–2(f) for the different values of F₂, which clearly demonstrate that an ultra-long acoustic jet can be achieved. In addition, the ray trajectories were calculated based on the ray tracing technique and overlayed with the simulated wave field in Figs. 2(a)–2(c), which exhibit good agreement with the numerical simulation results.

Furthermore, the acoustic intensity distributions of the double-foci MGLL are shown in Figs. 2(d)–2(f) and the normalized intensity profiles along the x and y axes (along the white dashed-dotted lines in Figs. 2(d)–2(f)) are plotted in Figs. 2(g) and 2(h). The jet length (JL) and FWHM of the acoustic jet were obtained based on the acoustic intensity along the x and y directions and are shown in Figs. 2(i) and 2(j). The JL is the length at which the intensity is above half of its maximum value along the x axis, which starts from the outer surface of the MGLL (x = 0.1 m). The FWHM is the length between the two locations at which the intensity is equal to half of its maximum value along the y axis. It can be clearly seen that the obtained acoustic jet has a JL over 17$λ$ at F₂ =3.2R, which starts from the outer surface of the Luneburg lens (near field) and ends at a distance of more than 17$λ$ from the outer surface of the lens (far field). In addition, the length of the acoustic jet can be tuned by changing the focal length F₂.

Perfect impedance-matching MGLL is an ideal case, which can hardly be manufactured in real applications. In this study, the GRIN acoustic metamaterials^23,24 are used to realize a practical double-foci MGLL for obtaining the acoustic jets. The acoustic metamaterial was designed to have the same geometric parameters as the impedance-matching double-foci Luneburg lens: R = 0.1 m, P_a = 0.5, F₁ = 1.2R, and three different values of F₂ (1.6R, 2.5R, and 3.2R). To obtain the refractive index distributions that are shown in Fig. 1(b), various unit cells can be used. Here, we used a unit cell of the three-dimensional (3D) lattice,^25,26 as shown in Fig. 3(a), which has three orthogonal beams such that each unit cell is interconnected with its adjacent cells to form a self-supported lattice. The unit cell used to build the two-dimensional (2D) lens is a 3D truss unit cell. The Luneburg lens made of such a 3D unit cell renders the lens to be stiff and free of deformation while being lightweight.

In the following studies, the periodicity of the unit cell D is chosen to be 5 mm. To determine the relation between the refractive index with the lattice unit cell, dispersion curves along the $ΓX$ direction based on the primitive cubic [as shown in Fig. 3(b)] were calculated by using the COMSOL software. The filling ratio of the unit cell can be changed through tailoring the factor a₀ and, hence, the dispersion curve can be changed correspondingly [see Fig. 3(c)]. The refractive index of each unit cell was calculated based on the slope of the dispersion curve.²⁷ Note that the slope of the dispersion curve is almost a constant value in the broadband frequency range of 0–20 kHz, which indicates that the refractive index is independent of the frequency and, therefore, the acoustic jet can be achieved in a broadband frequency range. Note that based on the homogeneity condition,²⁸ the working frequency is limited by the geometric parameters of the lens as 4D < λ < R. In this work, we designed the 2D double-foci Luneburg lens to work at the frequency range of f = 11–17 kHz with three different F₂ values (1.6R, 2.5R, and 3.2R), and an example of F₂ = 1.6R is shown in the inset of Fig. 3(c).

In this study, we explored the capability of the acoustic metamaterial double-foci Luneburg lens for producing an acoustic jet. The full 3D wave simulations were conducted using the commercial COMSOL software. Similar to the impedance-matching Luneburg lens simulations, the radiation boundary conditions were applied on the outer boundaries of the lens to assume the infinite air spaces. The dimension of the air area in the simulation was 7R × 4R. A plane wave of 17 kHz was used for the excitation at the location of x = −1.2R (y is from −2R to 2R). The numerical simulation results for both the waveform fields and intensity distributions are shown in Figs. 4(a)–4(f). The acoustic metamaterial MGLL is demonstrated to have the capability of generating an ultra-long acoustic jet, similar to that of the perfect impedance-matching Luneburg lens.

Similarly, the acoustic intensity profiles of the acoustic metamaterial double-foci MGLL along the x and y axes of the lens at the maximum intensity spots along the white dashed-dotted lines in Figs. 4(d)–4(f) are plotted and shown in Figs. 4(g) and 4(h). The JL and FWMH were obtained and shown in Figs. 4(i) and 4(j). Again, the acoustic metamaterial MGLL is proven to be able to achieve an ultra-long acoustic jet starting from the near field (outer surface of the lens) and extending to the far field (with JL more than 13$λ$ for F₂ =3.2R), and the FWHMs are around $λ$⁠.

To investigate the broadband characteristic of the acoustic metamaterial MGLL, the excitation signals of the different frequencies (f = 11, 13, and 15 kHz) were used for the MGLL with a focal length of F₂ = 2.5R. The simulated waveform fields and intensity distributions at different frequencies are shown in Fig. S1.²⁹ In addition, the acoustic intensity profiles of the acoustic metamaterial double-foci MGLL along the x direction and the corresponding JL were obtained and are shown in Fig. S2.²⁹ These results reveal the excellent performance of the MGLL for producing ultra-long acoustic jets in a broadband frequency range. The JL is higher at a lower frequency, and a maximum JL of over 30$λ$ can be obtained at 11 kHz. Note that the upper frequency range of the acoustic metamaterial double-foci MGLL scales linearly with the unit cell size without the change of JL/λ. Therefore, the lens can also be designed to work at higher frequencies by using smaller unit cell dimensions.

In addition, a quantitative comparison of the perfect impedance-matching MGLL and acoustic metamaterial MGLL was performed. The error in the jet length between the perfect impedance-matching MGLL and acoustic metamaterial MGLL is provided in Fig. S3,²⁹ which is calculated according to $Error=JLPL−JLAM/JLPL$⁠, where $JLPL$ is the jet length of the perfect impedance-matching MGLL and $JLAM$ is the jet length of the acoustic metamaterial MGLL. This error is due to the discrete refractive indices of the acoustic metamaterial MGLL, which generates impedance mismatch between the unit cells. As a result, part of the energy is reflected back and cannot propagate through the lens. For example, based on the simulations, there is almost no reflected energy for the perfect impedance-matching MGLL, whereas for the acoustic metamaterial MGLL with F₂ = 2.5R, around 15% of the energy is reflected back at the frequency of 17 kHz.

In this paper, we proposed a novel design of the GRIN acoustic metamaterial lens based on the MGLL to realize the double-foci, which can help achieve an ultra-long acoustic jet in a broadband frequency range. Theoretical studies and numerical simulations were performed to investigate the capability of the proposed lens for generating the ultra-long acoustic jet. The acoustic metamaterial MGLL was demonstrated to have a JL up to 30$λ$⁠, covering both the near field and far field.

This work was supported by the Air Force Office of Scientific Research (AFOSR) Center of Excellence on Nature-Inspired Flight Technologies and Ideas and United States Department of Agriculture (USDA) National Institute of Food and Agriculture (NIFA) Sustainable Agricultural Systems (Award No. 20206801231805).