Compact Acoustic Retroreflector Based on a Mirrored Luneburg Lens

PHYSICAL REVIEW MATERIALS 2, 105202 (2018)

Compact acoustic retroreﬂector based on a mirrored Luneburg lens

Yangyang Fu,1,2,3,4 Junfei Li,2 Yangbo Xie,2 Chen Shen,2 Yadong Xu,3 Huanyang Chen,4,* and Steven A. Cummer2,† 1College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China 2Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA 3School of Physical Science and Technology, Soochow University, No. 1 Shizi Street, Suzhou 215006, China 4Institute of Electromagnetics and Acoustics and Department of Electronic Science, Xiamen University, Xiamen 361005, China

(Received 24 September 2017; revised manuscript received 16 April 2018; published 18 October 2018)

We propose and demonstrate a compact acoustic retroreflector that reroutes incident signals back toward the source with minimal scattering. Gradient refractive index acoustic metamaterials, based on Archimedean spiral structures, are designed to fulfill the required refractive index profile. The experiments show that the compact acoustic retroreflector, whose radius is only approximately one wavelength, works in an incident angular range up to 120° over a bandwidth of about 27% of the central frequency. Such compact acoustic retroreflectors can be potentially applied in pulse-echo-based acoustic detection and communication, such as unmanned aerial vehicle sonar systems, robotic ranging detectors, and ultrasonic imaging systems.

DOI: 10.1103/PhysRevMaterials.2.105202

I. INTRODUCTION space-coiling metamaterials [26–28], which are either bulky or have limited working bandwidth, this spiral structure is A retroreflector is a volumetric or surface structure that nonresonant and has better impedance matching. Alterna- redirects the incident wave back toward the source, along tively, impedance-matched acoustic metamaterials could be the direction antiparallel to the direction of incidence. By realized using spatially dispersive techniques [29], which can rerouting the incident probing signal back toward the source exhibit extraordinary wide angle and broadband properties. or the transceiver, less signal is wasted in the uncontrolled However, these techniques [30,31] either work only for one- scattering process and higher signal-to-noise ratio (SNR) of dimension (1D) or introduce strong anisotropy, which is not the sensing signal is obtained. In optics, conventional op- appropriate for the Luneburg lens. tical retroreflectors, realized with corner reflector or cat’s By adding a reflected boundary on the focal surface of a eye configurations, can work within limited incident angles. Luneburg lens, waves can be directed back along the incident Alternatively, by employing transformation optics [1–3]to direction [as illustrated in Fig. 1(a)] with minimal lateral transmutate a refractive index singularity, an Eaton lens based shift. Compared with Eaton lens–based or metasurface-based on metamaterials [4–6] can be used as a retroreflector [7] retroreflectors [7,32], the scheme by combining a Luneb- to broaden the limited incident angle range. However, the urg lens and an arcual mirror releases the requirement of refractive index requirement for the Eaton lens–based retrore- high index profile or complicated geometrical configurations. flectors is extremely high. The resulting structures may have Different from the optical retroreflector in which the limit complicated geometries and high losses, which inevitably of polarization in three-dimensional (3D) design needs to hinder their real applications. be overcome, an acoustic Luneburg lens retroreflector is To overcome these difficulties, the use of a Luneburg lens easier to implement in the 3D case, which has significant [8–10] with gradient-index profile and a rigid back layer is value for practical applications. In addition, an acoustic proposed in this paper. Different from ray-based designs, the Luneburg lens retroreflector, generally, can be realized in gradient-index wave manipulation allows for a more compact acoustic microstructures where the effective refractive indices realization of the retroreflector. In airborne acoustics, by em- are obtained by tailoring effective path length. Therefore, ploying acoustic metamaterials [11–13] that can exhibit exotic the designed acoustic retroreflector can be easily extended to properties (e.g., negative mass density and bulk modulus) not higher frequency ranges by simply scaling the size according available in nature, the realization of Luneburg lenses has to the operating wavelength, which is not available in optical not been explored until recently, and these works [14–18] systems due to the dispersion of available materials. only focused on the focusing functionality of an acoustic To demonstrate the design concept, a compact acoustic Luneburg lens. In this paper, the discretized refractive in- retroreflector is realized with a radius of only approximately dex profile of a Luneburg lens is achieved with spiral unit one wavelength. The geometry-dependent properties also al- cells with tapered geometry [19–21]. Compared with cross- low the device to be conveniently scaled to function at various buck geometries [17,18], pillar type structures [22–25], and frequencies including airborne ultrasound. Both the numerical and the experimental results show our designed acoustic retroreflector can function within a relatively wide incident ◦ ◦ *[email protected] angle range (from –60 to +60 ) and over a bandwidth of †[email protected] about 27% of the central frequency.

FIG. 1. (a) The schematic of an acoustic retroreflector composed of a Luneburg lens (gradient color) and an arcual reflector (gray color). In plot, these curves are the propagation trajectory of the acoustic retroreflector. (b) The simulated total acoustic pressure field of an ideal acoustic ◦ retroreflector with a radius of R0 = 0.1 m for the incident beam with θ =−30 . The working frequency is 3.0 kHz and PML is the perfect matched layer. (c) The corresponding simulated acoustic pressure field patterns of the incident beam (the red arrow) toward the retroreflector and the reflected beam from the retroreflector (the blue arrow).

II. THEORETICAL MODEL AND NUMERICAL the retroreflector are shown, respectively, in Fig. 1(c).From VERIFICATION the information in Fig. 1(c), the reflected beam with planar wavefront, indeed, propagates along the direction parallel and The schematic of the acoustic retroreflector is shown in opposite to the incident one, with some scattered waves (see Fig. 1(a), where the retroreflector is composed of two ele- the reflected field outside the white frame) caused by the ments: a Luneburg lens (the refractive index is marked with wide incident beam toward the retroreflector (see the incident a color gradient) and an arcual mirror (marked with gray field outside the white frame). In fact, as the incident beam color). The working principle of the acoustic retroreflector is can converge at the interface between the Luneburg lens and displayed in Fig. 1(a), where the red and blue curves reveal the arcual area for a wide incident angle, accordingly such the propagation trajectory of the incident and reflected waves acoustic retroreflector can function in a wide incident angular in the acoustic retroreflector. Waves incident from the same range. Based on the above analysis and numerical results in direction [e.g., see the red solid and dashed curves in Fig. 1(a)] Fig. 1, the wide angle acoustic retroreflector could be obtained will converge at the same point locating at the surface of the using a mirrored Luneburg lens. Luneburg lens, thanks to the refractive index profile of the Luneburg lens given as n = 2 − (r/R )2, where R is the 0 0 III. DESIGN AND FABRICATION radius of the Luneburg lens. As the arcual area with sound- hard materials can function as a mirror, the converged incident To discretize such a continuous gradient-index profile re- waves will be reflected and return symmetrically back through quired, the designed Luneburg lens with R0 = 0.1m is di- the Luneburg lens. Consequently, the reflected waves will vided into five layers: a core with a radius of r0 = 0.02 m propagate along the direction that is parallel and opposite and four circular rings with thicknesses of 0.02 m. The to the incident one [see the blue solid and dashed curves effective refractive index of each layer is given as nl = 2 in Fig. 1(a)]. In a pulse-echo-based detection system, such 2 − [(l − 0.5)r0/R0] (l denotes the order of the divided retroreflection can significantly increase the signal-to-noise layers). Each layer is composed of unit cells with the ef- ratio of the echo signal and thus improve the sensing perfor- fective refractive indices realized by acoustic metamaterials. mance (e.g., longer detection range). Therefore, five different kinds of acoustic metamaterials are To verify this concept, the corresponding simulation with designed for these unit cells. The core region is filled with COMSOL MULTIPHYSICS is shown in Fig. 1(b), where the a single unit cell and its size is 0.04 m × 0.04 m. The unit proposed acoustic retroreflector is comprised of an ideal cells in the four circular ring regions are the same sizes, i.e., Luneburg lens (R0 = 0.1 m) and an arcual ring (its central 0.02 m × 0.02 m, and as a result there are successively 6, ◦ angle is 150 ; more discussion for the central angle is shown 12, 24, and 36 unit cells from the inner circular ring to the in the Supplemental Material [33]) of a sound-hard material. outer circular ring. Although many topologies can be adopted ◦ For the incident beam with an incident angle of θ =−30 for the unit cell design with discrete refractive index, here (the incident angle is defined as the deviation angle from the we choose tapered spiral cells as the candidate. Compared geometric symmetric axis of the retroreflector, with the left with structures such as crossbuck structure [17,18], pillars deviation angle defined as “–”), there is enhanced pressure [22–25], or perforated plates [34,35], the spiral cells we field in the incident area where the incident wave can enter used here have simple geometrical configurations and better into the Luneburg lens (see the white dashed frame), implying impedance matching, and are expected to have relatively low that the propagation direction of the reflected wave (the blue loss. To illustrate the design details of acoustic metamaterials arrow) is parallel and opposite to the incident one (the red in these unit cells, here we take the unit cell for obtaining arrow). To display the effect of the acoustic retroreflector n3 = 1.33 (l = 3) as an example shown in Fig. 2(a), where more clearly, the acoustic fields of the incident beam to- the unit cell is composed of four identical microstructures ward the retroreflector and the reflected beam returning from possessing fourfold rotational symmetry. The thickness of the

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FIG. 2. (a) The design details of the spiral acoustic metamaterials. (b) The designed Luneburg lens working at 3.0 kHz with discrete index profiles realized by gradient acoustic metamaterials. (c) Effective refractive index of the unit cells as a function of frequency. (d) The schematic of the experimental setup for the acoustic retroreflector. Inset shows the fabricated simple of the acoustic retroreflector working at 3.0 kHz. wall is t = 0.001 m, with its geometrical shape described by subwavelength unit cells. In addition, to show the dispersion the Archimedean spiral [19], where quantitative analysis of of the unit cells, the effective refractive index is also plotted the impedance of the spiral structure has been reported. By as a function of frequency, as depicted in Fig. 2(c). Within designing the appropriate parameters for the Archimedean the frequency range from 2.5 to 3.5 kHz, the effective indices spiral, the required effective refractive index of this unit cell remain roughly the same. can be obtained. The effective refractive index is retrieved Furthermore, based on the design of the Luneburg lens, the from the transmission and reflection coefficients of the unit acoustic retroreflector working at 3.0 kHz is fabricated using cell in a waveguide [36] [see the simulated field pattern in fused filament 3D printing technology [see inset of Fig. 2(d)]. Fig. 2(a)]. Accordingly, by designing five different acoustic To test the performance of the acoustic retroreflector, the metamaterial unit cells, the Luneburg lens working at 3.0 kHz experimental setup is schematically shown in Fig. 2(d).The was designed as shown in Fig. 2(b), where these different retroreflector is situated in a two-dimensional waveguide. The acoustic metamaterials are, respectively, placed in the cor- incident wave is a Gaussian modulated plane wave generated responding areas to realize these particular indices marked by the speaker array with its propagation direction normal to by different colors. These unit cells with fourfold rotational the flat speaker array. The acoustic field is scanned with a symmetry possess essentially isotropic material properties, moving microphone with a step size of 2 cm. The received which can be deduced from circle dispersion relationships signals are then time windowed to extract the incident and (Supplemental Material [33]). Although these designed unit reflected waves. A Fourier transform is then performed to cells have slight geometric deformation with the curved sec- obtain the acoustic field information at different frequencies. tors and the central circle, the sizes of the unit cells are To measure the sample response at different incident angles, much smaller than the working wavelength, and therefore the sample is rotated at the corresponding angle while keeping the geometric deformations are assumed to have negligible the incident beam unchanged. effect on the effective indices. To demonstrate the performance of the designed Luneburg lens, the simulated acoustic IV. EXPERIMENTAL DEMONSTRATION field patterns for several incident angles are displayed in the Supplemental Material [33]. These simulated field patterns A. Near-field performance are almost identical and are also consistent with that from Figures 3(a) and 3(b) show the simulated total pressure the ideal Luneburg lens, implying that the designed Luneburg field and the simulated and measured reflected field for a beam lens is angle independent, and that the ideal continuously incident on the designed retroreflector at θ = 0◦.Fromthe varying index profile is well presented by our discretized simulation in Fig. 3(a), the total pressure field is enhanced in

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FIG. 3. (a,b) are the simulated total pressure field pattern and the simulated (left) and the measured (right) reflected pressure field patterns from the retroreflector for an incident beam at θ = 0◦ on the designed retroreflector at 3.0 kHz. (c,d) are the simulated total pressure field pattern and the simulated (left) and the measured (right) reflected pressure field patterns for an incident beam with θ = 0◦ on a cylinder with R0 = 0.1m. the incident area (see the white dashed frame), which means from a cylinder. Although the reflected near field from the that the reflected wave (the blue arrow) propagates toward retroreflector is not stronger than that from the cylinder, it the incident one (the red arrow). To observe the performance is caused by loss in the designed device. If the reflected of the designed acoustic retroreflector clearly, the simulated wave is measured in the far field, the reflected wave with a reflected pressure field is shown in the left part of Fig. 3(b), planar wavefront from the retroreflector will be enhanced in where a flattened wavefront of a Luneburg lens is seen in the comparison to that from a cylinder. retroreflector. Particularly, the reflected beam in the incident Similar retroreflection performance is seen in Fig. 4 for the area marked by the white dashed frame possesses a planar incident angles with θ =−30◦ and θ =−60◦.Forθ =−30◦, wavefront, implying that the incident beam is directionally the enhanced total pressure field is seen in Fig. 4 (a) and the retroreflected and not broadly scattered. simulated and measured reflected fields consistently reveal The corresponding measured reflected pressure field, the planar wavefront in the incident area [see Fig. 4(b)]. For marked by the black dashed frame, is shown in the right θ =−60◦, as a part of the focusing region of the incident side of Fig. 3(b); it shows that the experimental result is in wave is beyond the arcual reflector, some incident wave good agreement with the simulated one (the simulated and penetrates through the retroreflector [see Fig. 4(c)]. As a measured results share the identical color bar). Although the result, the intensity of the reflected wave is reduced and the wavefront is curved outside of the incident area caused by performance of the planar wavefront is decreased, caused by the arcual reflector, the wavefront of the reflected field in the more scattering from the arcual reflector [see Fig. 4(d)]. As the incident area, indeed, is planar. To highlight this feature of designed acoustic retroreflector possesses a symmetric index our designed retroreflector, the corresponding results of an profile, the designed acoustic retroreflector can function for a ◦ ◦ ◦ incident beam with θ = 0 on a cylinder with R0 = 0.1m wide incident angle, with a range from θ =−60 to θ = 60 . are shown in Figs. 3(c) and 3(d). There is an enhanced Accordingly, wide angle acoustic retroreflector performance total pressure field in the incident area, but also much more at 3.0 kHz is clearly demonstrated in both simulated and ex- wave scattering [see Fig. 3(c)]. By observing the simulated perimental results. In addition, the wide angle designed acous- reflected field pattern in the left image of Fig. 3(d), there is tic retroreflector also performs well at other frequencies, as a monopolelike, omnidirectional cylindrical radiation. This is shown in the corresponding simulated and measured reflected confirmed by the measured result in the right plot of Fig. 3(d). waves at 2.6 and 3.4 kHz (see Supplemental Material [33]). Comparing the simulated reflected fields in Fig. 3,wesee that the reflected field from the retroreflector is stronger than that from a cylinder. This is because the reflected wave from B. Far-field performance the proposed retroreflector in the incident area is unattenu- We now extrapolate from the near-field measurements and ated radiation with a planar wavefront, while it is attenuated simulations to determine the far-field performance of this radiation with a cylindrical wavefront for the reflected wave acoustic retroreflector. As an illustration, the reflected signal

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FIG. 4. (a,b) are the simulated total pressure field pattern and the simulated (left) and the measured (right) reflected pressure field patterns from the retroreflector for the incident beams with θ =−30◦ bumping on the designed retroreflector at 3.0 kHz, respectively. (c,d) are the simulated total pressure field pattern and the simulated (left) and the measured (right) reflected pressure field patterns from the retroreflector for the incident beams with θ =−60◦ bumping on the retroreflector at 3.0 kHz, respectively. in the far field for normal incidence (θ = 0◦)isshownin Relative to the retroreflected power from a cylinder, there Fig. 5, where the normalized reflected power at a distance is nearly three times more retroreflected power from the of D = 50λ is numerically and experimentally revealed in a designed retroreflector in experiment. To further illustrate polar pattern. In both the numerical and experimental plots, the effect of losses, the Supplemental Material [33] reveals the reflected power at D = 50λ in the direction of 0◦ for performance simulations for the designed retroreflector with ◦ a beam incident on a R0 = 0.1 m cylinder with θ = 0 is different levels of loss. We can see that while the reflected normalized as unity. energy from the retroreflector decreases with increasing loss, We first focus on the numerical results. From Fig. 5(a),the the basic function of retroreflection is well preserved, as the reflected energy from the cylinder is almost uniform in all maximum reflected energy occurs in the direction opposite to directions, caused by the cylindrical radiation (see the black incident direction. dotted curve). In contrast, for the designed retroreflector at We performed simulations for normal incidence from 2.0 3.0 kHz [see the red solid curve in Fig. 5(a)], the highest to 4.0 kHz to assess the device bandwidth. Performance is reflected power is observed in the direction of 0◦. For this expected to drop at lower frequencies because the device case the retroreflected power from the retroreflector is four becomes smaller relative to a wavelength, and also at higher times that from the cylinder. There is enhanced power in frequencies because the discretized Luneburg approximation the 30◦ direction in the simulation [see the red solid curve begins to break down. We find that good retroreflection perfor- in Fig. 5(a)], and also a weak enhancement in the −30◦ mance extends to the range between 2.6 and 3.4 kHz, which direction. Simulations of the reflection at normal incidence is shown by the blue and green curves in Fig. 5(b).Theex- from the arcual reflector by itself [see the gray dashed curve perimental results agree well with the simulations. Therefore, in Fig. 5(a)] show significant reflection of −30◦ and 30◦. the enhanced and directional reflection is well demonstrated We thus attribute the reflected power in these two directions in our designed acoustic retroreflector, functioning in a 27% to the arcual reflector of the retroreflector. The asymmetry bandwidth about the central design frequency. of reflection in these two directions is caused by the spiral As the designed retroreflector can function in a wide inci- microstructures in the retroreflector. dent range, similar results can be achieved for other incident The experimental measurements confirm the good ex- angles; e.g., see the case of θ =−30◦ in the Supplemental pected retroreflector performance. The experimental result for Material [33]. In fact, even if the retroreflector is not placed a normally incident beam on the retroreflector at 3.0 kHz in the central direction of the incident beam, e.g., it deviates (see the red dot-dashed curve) is also shown in Fig. 5(a). from the central axis with a distance of one wavelength, the Here the far-field date is obtained through the projection of enhanced retroreflected signal can be observed as well [33]. It the measured near field based on the Huygens principle. There is also worth noting that our retroreflector’s radius is approx- is less reflected power in the direction of 0◦ in experiment imately equal to one wavelength. However, if the size of our caused by the loss in the designed retroreflector, but the proposed retroreflector is increased [33], its performance will experimental and numerical results are in good agreement. improve further.

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120° over a bandwidth of about 27% of the central frequency. In addition, we have also performed transient simulations and experiments to study the pulse-echo-based performance of the retroreflector [see Supplemental Material [33], Videos 1, 2, 3 (simulation), and 4 (experiment), for the full time evolution of the pressure fields], and the results are consistent with the frequency domain simulation. The detailed setup and results are summarized in the Supplemental Material [33]. Also, since the spiral unit cells are not axially symmetric, there will be some bias of the reflected field created by this kind of asymmetry. This can be improved with more unit cells so that the effect of cell asymmetry can be lowered. As the refractive index is controlled solely by the unit cell geometry, the device may be conveniently scaled with respect to the operation frequency, including airborne ultrasound for most commercial sonar systems. We also carry out numerical simulations for a designed retroreflector working in 30 kHz, shown in the Supplemental Material [33], in which the performance of the designed retroreflector is maintained even with a larger viscosity. In addition, the performance of the proposed retroreflector might be well optimized using other acoustic metamaterials [37,38] and also could be extended to three dimensions [39,40]. We expect our proposed retroreflector to be potentially useful for pulse-echo-based acoustic FIG. 5. (a) The polar pattern of the normalized reflected power sensing and communication, and is expected to be of interest ◦ at D = 50λ for a normally incident beam (θ = 0 ) on different to researchers in areas such as robotic sensing, ultrasonic structures. The gray dashed and black dotted curves are the cases imaging, and distributed sensor networks. For example, such of the reflected power from the arcual ring and cylinder at 3.0 retroreflectors could be potentially used to improve point- kHz, respectively, and the red solid and red dot-dashed curves are to-point communication among such distributed networks, the cases of the numerically and experimentally reflected power and the enhanced reflected signal will only propagate in the from the designed retroreflector at 3.0 kHz, respectively. (b) The direction where it comes from, which can help them to avoid polar pattern of the normalized reflected power at D = 50λ for the ◦ surveillance outside the network. normally incident beam (θ = 0 ) on the designed retroreflector at other frequencies. The blue dashed (green solid) and blue dot-dot- dashed (green dot-dashed) curves are the cases of the numerically ACKNOWLEDGMENTS and experimentally reflected power from the designed retroreflector at 2.6 kHz (3.4 kHz), respectively. J.L., Y.X., C.S., and S.A.C. were supported by a Multidis- ciplinary University Research Initiative grant from the Office of Naval Research (Grant No. N00014–13-1–0631) and an V. CONCLUSIONS Emerging Frontiers in Research and Innovation grant from the In conclusion, we have designed a compact acoustic mir- US National Science Foundation (Grant No. 16-41084). Y.X. rored Luneburg lens by placing a rigid arcual back adjacent to was supported by the National Natural Science Foundation of the original lens. The device greatly expands the functionality China (NSFC) (Grant No. 11604229), and H.C. was supported of the Luneburg lens and can perform as an acoustic retrore- by the National Science Foundation of China for Excellent flector. The retroreflector, with a radius of only approximately Young Scientists (Grant No. 61322504). one wavelength, works in an incident angular range up to Y.F., J.L., and Y. Xie contributed equally to this work.

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