A conventional structural Luneburg lens is a symmetric circular gradient-index lens with refractive indices decreasing from the centre along the radial direction. In this paper, a flattened structural Luneburg lens (FSLL) based on structural thickness variations is designed by using the quasi-conformal transformation technique. Through numerical simulations and experimental studies, the FSLL is demonstrated to have excellent beam steering performance for the manipulation of flexural wave propagation at desired angles.
Structural beamforming is a critical technique in the areas of structural health monitoring and non-destructive testing and has attracted widespread interest.1,2 To achieve multi-directional beamforming, conventional phased array techniques based on phase-shift systems are employed, which often require complicated design, expensive fabrication, and high input power.3,4 With the development of graded-index metamaterials in recent years,5–13 Luneburg lenses are receiving considerable attention for the manipulation of not only optical waves14,15 and acoustic waves,16–18 but also structural waves.19–21 Compared with conventional phased array techniques, the structural Luneburg lens, as an attractive graded-index lens, can be used to achieve arbitrary directional beamforming without the need for any phase-shift systems.22,23
The conventional structural Luneburg lens is a circular gradient-index lens and every point on the lens surface can act as a source location for generating plane wave propagation on the opposite side of the lens. However, the applications of the conventional structural Luneburg lens are limited due to its circular aperture. The lens can hardly accommodate flat feeding sources used in practical applications. Furthermore, as the lens is often an integral part of the test structure, the circular aperture requires the placement of transducers inside the structure, which is not always feasible.
In this study, we propose a flattened structural Luneburg lens (FSLL) for flexural wave beamforming. The FSLL is designed by using the quasi-conformal transformation (QCT) technique, which allows one to modify the structure geometry and refractive index distribution with minimum anisotropy and reduced computational complexity.24–27 The QCT technique has been explored in the design of optical and acoustic metamaterial structures.28–33 The design principle of our FSLL based on the QCT is illustrated in Fig. 1. The original circular structural Luneburg lens, denoted as the virtual space in Fig. 1(a), is transformed into an FSLL denoted as the physical space in Fig. 1(b). Feeding sources (e.g., a line source, several individual sources, or a moving single source) can be attached to the flat side of the lens to achieve beamforming of different angles. The refractive index distribution of the FSLL in the physical space is calculated by solving Laplace's equation in the virtual space [Fig. 1(a)] with Dirichlet and Neumann boundary conditions given below,34
where (x, y) and (x′, y′) are the coordinates of the virtual space and the physical space, respectively, represents the outward normal vector to the surface boundaries, and is the gradient operator.
We then obtain the effective refractive index of the FSLL as follows:35,36
where n and n0 represent the refractive indices of the circular structural Luneburg lens and the background medium, respectively, and J is the Jacobian matrix.
The commercial software comsol is used to solve the Laplace equation with the corresponding boundary conditions to obtain the refractive index distribution of the FSLL. The normalized refractive index distribution (n/n0) of the virtual space is shown in Fig. 1(c). The transformed shape and normalized refractive index distribution (n′/n0) of the physical space are shown in Fig. 1(d).
The FSLL is constructed by using a variable thickness structure defined in a thin plate. Based on the obtained refractive index distribution of the FSLL in Fig. 1(d), the range of the refractive index is between 1 and 1.82. This refractive index variation can be realized by varying the structural thickness. The same approach has been employed for achieving acoustic black holes37–40 and other types of structural lenses.23,41 By using Eq. (2), a graded variation of thickness h can be obtained as23
where h0 is the constant plate thickness.
Note that our proposed FSLL is a broadband device in principle. There is no upper bound frequency for the FSLL due to its continuously varying refractive indices. On the other hand, the lower bound frequency is limited by the scattering regime, which requires the wavelength λ to be less than the radius of the lens R.
For proof-of-concept, the following parameters are chosen in the following studies: radius R = 5 cm, open angle φ = 90°, constant thin plate thickness h0 = 0.004 m, and dimension of the entire plate l0 × w0 × h0 = 0.45 m × 0.3 m × 0.004 m. The open angle defines the azimuthal angle range of the beamforming, which can be designed to be any value from 0° to 180°. Here, for the designed open angle of 90°, the beam can be steered over the azimuthal angle range from −45° to 45°, when a feeding source moves along the flat side of the lens. For simplicity without losing the generality, two feeding source locations were considered, as shown in Fig. 1(b). They are labelled as 1 (x′ = 0 mm, y′ = 0 mm) and 2 (x′ = −8 mm, y′ = 0 mm), which correspond to the steering angles of approximately 0° and 16.5°, respectively.
Full 3D wave simulations were conducted by using commercial software comsol. Both frequency and time domain analyses were carried out to show the flexural wave beamforming by using the FSLL. Perfectly matched layers (PMLs) and low reflecting boundaries (LRBs) were used to reduce the boundary reflections in frequency response and time domain analysis correspondingly. The frequency domain analysis was performed for the frequency range from 20 to 80 kHz. Based on the lens radius R = 5 cm, we chose the lower bound frequency to be 20 kHz. Although there is no theoretical upper bound frequency, due to the limitation of the numerical simulations, which require 5–10 elements for simulating the smallest wavelength, an upper bound frequency of 80 kHz was chosen in this study.
The obtained waveforms at a frequency of 40 kHz were shown in Figs. 2(a) and 2(b). It can be seen that the main wave energy propagates along the steering angles of 0° and 16.5° (the black dot dashed rectangles), respectively, when point sources 1 and 2 are excited. The phase plots are presented in Figs. 2(c) and 2(d), which clearly show that the circular phase originated from the source is transformed into flat phases along the steering angle directions (the black dot dashed rectangles). These results demonstrate that the outgoing waves are collimated at steering angles of 0° and 16.5°. Simulation results obtained for beam steering angles of 30° and 45° are provided in Fig. S1 and Fig. S2 of the supplementary material,42 respectively. In order to characterize the broadband beamforming performance of the FSLL, the phase data along the direction that is perpendicular to the beamforming direction [e.g., white dashed lines in Figs. 2(c) and 2(d)] are plotted in Figs. 2(e) and 2(f) over a broad frequency range of 20–80 kHz. These results clearly show that the phase data obtained for the outgoing beams along the corresponding steering angle directions are constant over the entire simulated frequency range, demonstrating the broadband beamforming characteristic of the FSLL.
Furthermore, a time domain analysis was performed to examine the flexural wave beam steering at different time steps. A signal of three-count tone bursts with a central frequency of 40 kHz was used as the excitation source. The full field waveforms of propagation at different time steps obtained for the steering angles of 0° and 16.5° are shown in Figs. 3(a)–3(c) and Figs. 3(d)–3(f), respectively. Again, the main wave energy is shown to propagate along the directions of the steering angles (0° and 16.5°). From t = 0 to 0.05 ms, the circular flexural waves originated from the point source propagate forward. At t = 0.1 ms, the flexural waves interact with the FSLL and the waveforms gradually become collimated. After t = 0.15 ms, the waves propagate forward as plane waves.
Experimental studies were also carried out to characterize the performance of an FSLL fabricated on a thin plate (6061 aluminium from McMaster-Carr) with dimensions of 0.6 m × 0.3 m × 0.004 m. Note that the minimum thickness of the fabricated FSLL is only 1.2 mm, which could subject to damage due to high power excitations. However, this thin structure is still applicable for structure health monitoring based on Lamb waves, in which the displacement under low power excitation of PZT transducers is usually in the range of nanometers to micrometers.43,44 The experimental setup and the fabricated FSLL are shown in Fig. 4. The four sides of the plate were covered with plastilina modelling clay to minimize boundary reflections. Two circular piezoelectric discs (PZT) (12 mm in diameter and 0.6 mm in thickness from STEMiNC Corp.) were used as point sources, which were placed at the two feeding source locations [1 (x′ = 0 mm, y′ = 0 mm) and 2 (x′ = −8 mm, y′ = 0 mm)] shown in Fig. 1(b). A piezoelectric transducer was bonded to the plate by using an adhesive (2 P-10 adhesive from Fastcap, LLC). A scanning laser Doppler vibrometer (SLDV) (PSV-400 from Polytec) was used to measure the propagating wave field by recording the out-of-plane particle velocity on the plate as shown in Fig. 4(a). The scanning was performed on the front surface (flat side) of the plate, as shown in Fig. 4(c). The back surface of the plate with the FSLL is shown in Fig. 4(b). A detailed description of the experimental setup is provided in the supplementary material42 with a schematic of the fabricated plate shown in Fig. S3. In the experiment, a voltage signal of three-count tone bursts with a central frequency of 40 kHz was used to excite the transducer. The input signal (see Fig. S4 in the supplementary material42) contains frequency information from 20 to 60 kHz and the 3DB bandwidth is 28 kHz (from 26 to 54 kHz). Transient responses were obtained to examine the beamforming performance of the FSLL. Note that only a limited frequency range defined by the input three-count tone bursts was used in the experiment. For higher frequencies, the strong reflections from the plate boundary can obfuscate the performance of beamforming. On the other hand, the experimental studies at lower frequencies would require a much larger plate, which causes unnecessary complication for the experiment. Therefore, we choose the input signal at a central frequency of 40 kHz with a 3DB bandwidth of 28 kHz.
The measured waveforms of the out-of-plane particle velocity fields at different time instants for two steering angles of 0° and 16.5° are shown in Figs. 4(d)–4(f) and Figs. 4(g)–4(i), respectively. These experimental results are in excellent agreement with the numerical simulation results shown in Fig. 3, which further validate the beam steering capability of the FSLL.
We numerically and experimentally demonstrated a FSLL for beam steering of flexural waves. The FSLL was designed by using the QCT technique with Dirichlet and Neumann boundary conditions. The flattened lens surface allows easy accommodation of feeding sources for practical applications. In this study, the FSLL was fabricated by using a variable thickness structure defined in a thin plate, which rendered a continuous change of the refractive index for smooth manipulation of flexural wave propagation. The numerical simulation results indicate that the FSLL can successfully perform beamforming at the designed steering angles over a broad frequency range. The experimental results further validated the beam steering performance of the FSLL at a particular frequency, which were in good agreement with the numerical simulation results. The FSLL based beamforming could benefit many potential applications including ultrasonic imaging, structural health monitoring, and non-destructive testing.
Partial financial support from Khalifa University on the Pipeline System Integrity Management (PSIM) project is gratefully acknowledged.