Transparent gradient index lens for underwater sound based on phase advance
Theodore P. Martin, Christina J. Naify, Elizabeth A. Skerritt, Christopher N. Layman, Michael Nicholas, David C. Calvo, Gregory J. Orris, Daniel Torrent, Jose Sanchez-Dehesa
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Transparent gradient index lens for underwater sound based on phase advance
Theodore P. Martin, ∗ Christina J. Naify, Elizabeth A. Skerritt, ChristopherN. Layman, Michael Nicholas, David C. Calvo, and Gregory J. Orris
U.S. Naval Research Laboratory, Code 7160, Washington, DC 20375, USA
Daniel Torrent and Jos´e S´anchez-Dehesa
Department of Electronic Engineering, Wave Phenomena Group,Universidad Politecnica de Valencia, C/ Camino de Vera s7n, E-46002 Valencia, Spain (Dated: August 30, 2018)Spatial gradients in refractive index have been used extensively in acoustic metamaterial applica-tions to control wave propagation through phase delay. This study reports the design and experi-mental realization of an acoustic gradient index lens using a sonic crystal lattice that is impedancematched to water over a broad bandwidth. In contrast to previous designs, the underlying latticefeatures refractive indices that are lower than the water background, which facilitates propagationcontrol based on a phase advance as opposed to a delay. The index gradient is achieved by varyingthe filling fraction of hollow, air-filled aluminum tubes that individually exhibit a higher sound speedthan water and matched impedance. Acoustic focusing is observed over a broad bandwidth of fre-quencies in the homogenization limit of the lattice, with intensity magnifications in excess of 7 dB.An anisotropic lattice design facilitates a flat-faceted geometry with low backscattering at 18 dBbelow the incident sound pressure level. Three dimensional Rayleigh-Sommerfeld integration thataccounts for the anisotropic refraction is used to accurately predict the experimentally measuredfocal patterns.
PACS numbers: 43.20.+g,43.30+m,43.40.+s
Metamaterial lattices composed of sub-wavelength scattering components have been used increasingly in recentyears to control the propagation of both electromagnetic and acoustic waves through two- and three-dimensions. Oneprimary goal of acoustic metamaterial design has been to achieve effective fluid, or “metafluid,” material propertiesthat minimize shear coupling and propagation. The coupling between a fluid-born acoustic wave and a compositeelastic material at off-normal incidence results in mode-mixing of dilatation and shear modes that complicates theinteraction. For underwater applications requiring strong acoustic coupling, compliant materials such as rubbers havebeen a standard material used for coupling and encapsulation due to their relatively close impedance match with waterand low shear moduli. Traditional compliant materials often have larger densities and/or lower dilatation moduli thanwater, making them useful for applications requiring low relative sound speed. For example, a sonic crystal lattice ofrubber cylinders with a gradient filling fraction was recently used to achieve an underwater omnidirectional absorptioncoating [1], which requires a radially decreasing sound speed. [2, 3]Metafluids with complementary material properties compared to rubbers, e.g. high relative sound speeds withmatched impedance, have been more difficult to achieve in underwater applications because they require high stiffnessbut low relative density. Sonic crystal lattices feature metafluid functionality in the homogenization limit of the lattice,including shear decoupling and broadband performance, and hence they have the potential to expand the range ofrealizable metafluid material properties to include high sound speeds. Thin-walled, hollow elastic shells have beenrecently proposed as high-sound speed sonic crystal components in water. [4] By carefully tuning the wall thicknessof the shell, an impedance-matched condition can be obtained assuming the shell material has higher impedance thanthe background fluid. Additionally, by using shell elements which are individually impedance matched, it has beenshown that any phononic crystal configuration of the elements will also be impedance matched in the homogenizationlimit of the crystal lattice. [4] Given that local variations in the element filling fraction create a variation in refractiveindex, acoustically transparent devices with tunable wave-guiding capability should be achievable using hollow shellelements. The concept of hollow shell elements has also been extended to include structural components inside theshells to improve control over the material properties.[5]Here we report the design and experimental demonstration of an acoustically transparent gradient index (GRIN) lensthat focuses sound by advancing the phase of waves propagating through an aqueous background. GRIN geometrieshave been utilized for a broad range of metamaterial applications including scattering reduction [6, 7], wave focusing [8–16] and bending [1–3, 17, 18]. In contrast to previous lens designs, we present a lens composed of impedance-matched, ∗ Electronic address: [email protected] hollow shell elements with sound speeds higher than water. The higher sound speed enables the phase of propagatingwaves to be advanced beyond what is possible in the background propagation medium, which is important to a numberof metamaterial applications (for a review, see Refs. 19, 20). The lens is intended as a proof of concept to demonstratethat both transparency and broadband wave-guiding functionality can be achieved simultaneously using a lattice ofnon-compliant, low refractive index scattering elements.The GRIN lens is constructed using cylindrical, air-filled aluminum tubes arranged in a lattice that function asa broadband, uniform effective medium at wavelengths λ larger than the lattice spacing a ( λ > a ). With a wallthickness of 1 / a y compared to the lattice constant a x in the orthogonal direction. Phasespeeds ¯ c xx and ¯ c yy in the primary Cartesian directions are shown in Fig. 1(b) as a function of the anisotropy ratio a y /a x for a fixed aluminum tube outer radius R = 0 . a x . The double index on the phase speeds delineates componentsin an anisotropic tensor. Phase speeds ¯ c with an over-bar are normalized to the water background, which is assumedto have density ρ b = 1000 kg / m and sound speed c b = 1480 m / s. The phase speeds are derived from the longitudinaldispersion bands of the acoustic band structure calculated for each anisotropy ratio. The anisotropy in phase speedcan also be derived by considering multiple scattering effects in the lattice. [21]Examples of the acoustic band structure are shown in Fig. 1(c,d). Blue and red lines, with slopes correspondingto the phase speeds in Fig. 1(b), demonstrate that the longitudinal bands have linear dispersion up to an uppercutoff frequency ωa x / πc b ≃ .
3. The cutoff frequency is constrained by the wrapping of the longitudinal band atthe Brillioun zone boundary in the Γ Y direction for the highest anisotropy ratio considered. The longitudinal bandin the Γ M direction in Fig. 1(c) has a slowness that deviates slightly from circularity due to the underlying cubicsymmetry. In addition to the longitudinal bands, narrow resonance bands arising from the core-shell architecture ofthe unit cell are predicted at various frequencies.[5] While we do not observe evidence of coupling to these bands inour measurements, the resonant bands should not, in general, be overlooked depending on the lattice geometry.Our lens design consists of a constant lattice spacing in the x -direction, which produces a flat lens facet, while aprescribed lattice anisotropy in the y -direction produces the convex GRIN profile [see Fig. 2(b)]. The primary impactof increasing the anisotropy ratio is to decrease the effective sound speed irrespective of propagation direction. Asis evident in Fig. 1(b), the difference between ¯ c xx and ¯ c yy is small even at a y /a x = 2, resulting in approximatelyisotropic acoustic transport over the entire Brillioun zone. Therefore, an approximate, isotropic sound speed ¯ c avg =(¯ c xx + ¯ c yy ) / n ( y ) = 1[¯ c avg ( y )] = β ¯ c − ( α y ) + (1 − β ) ¯ c (cosh α y ) ! − (1) α = (2 /h ) q − (¯ c / ¯ c h ) (2) α = (2 /h ) arccosh (¯ c h / ¯ c ) (3)where h is the lens height in the gradient direction, ¯ c = ¯ c avg (0) and ¯ c h = ¯ c avg ( h/
2) are the extremal sound speedsobtained from the chosen range of a y /a x , and β is a parameter that mixes the two GRIN profiles.An iterative ray tracing routine based on the eikonal approximation [22] was used to determine the optimal valueof β that minimizes the lens aberration. The isotropic refraction gradient in Eq. (1) prescribes ¯ c avg ( y ) and hencethe lattice geometry a y /a x for a given value of β ; however, to optimize β an eikonal approximation that accountsfor the anisotropic sound speeds in the lattice was employed to improve the accuracy of the physical geometry. Thedifferential equation governing the eikonal function ξ ( x ) in the presence of anisotropy can be derived for acousticmetafluids in a similar manner to the electromagnetic case, [22]¯ c xx ξ x + ¯ c yy ξ y − c xy ξ x ξ y = 1 (4)where ξ [ x,y ] = dξ/d [ x, y ]. The off-diagonal sound speed ¯ c xy can be calculated from the band structure or by consideringmultiple scattering effects. [21] The off-diagonal term was found to be negligible compared to the on-diagonal termsfor each anisotropy ratio in our lattice.Figure 2(a) shows the ray paths derived using Eq. (4) for a GRIN lens insonified by a plane wave after optimizingthe parameter β . The lens has thickness 10 a x , height h = 27 . a x , and lattice spacing a x = 15 . β = 0 .
65 produces a focal point at x ≃ . β -optimization routine starts with the averagesound speed profile ¯ c avg ( y ) as input and converts to the anisotropic phase speeds from Fig. 1(b). The optimizationroutine also ensures that the gradient in the lattice spacing a y ( y ) is properly discretized to be consistent with thetotal height h of the optimized GRIN profile.A schematic of the upper half of the β -optimized lens design is shown in Fig. 2(b). The full lens is symmetricabout the y -axis. A photograph of the assembled lens is shown in Fig. 2(c). The lens is constructed of identical 1 mlong hollow aluminum cylinders with air inside. Each cylindrical tube has a diameter of 12.5 mm and nominal wallthickness of 0.625 mm. Rubber end caps are adhered to both ends of each cylinder to prevent water infiltration.Additionally, ∼ . a y = a x are placed at the top and bottom of the lens to help mitigate the sharp transitionwhere the edge of the lens (¯ c avg = 1 .
36) meets the open water (¯ c avg = 1).The lens was submerged at the center of a 6 × × water tank. Acoustic waves were produced by a 0.1 mdiameter spherical source located in the plane that bisects the lens midpoint in the axial direction ( z -axis). In orderto demonstrate the lens directionality, two in-plane source locations were considered at ( x, y ) = ( − . ,
0) m and( x, y ) = ( − . , .
43) m corresponding to incident angles of 0 ◦ and 15 ◦ respectively from the lens central axis ( x -axis).Wave propagation was measured using hydrophones at a sampling rate of 0.8 MHz. Hydrophones were mounted on athree-axis translation positioning system to record measurements of the sound pressure levels, P ( x, y ) and P ( x, y ), inthe presence and absence of the lens respectively. Measurements were taken at 10 mm increments. The sound pressurelevel was measured at individual frequencies by averaging over a 10-cycle pulse in the transmission region (as indicatedin Figs. 3 and 4) and a 20-cycle pulse in the reflection region. The longer pulse cycle used in the reflection regionensured overlap between incident and reflected waves so that the measured total field could be properly comparedwith simulations. The length of the incident pulses were short enough to isolate reflections from the tank walls.Measurements were performed in the xy -plane perpendicular to the cylindrical axis of the lens over a broad rangeof frequencies ωa x / πc b < .
3. Examples of the measured and numerically modeled pressure intensity ( | P | / | P | ) inthe vicinity of the lens are shown in Figs. 3 and 4 for frequencies ωa x / πc b = 0 . y -axis, measurements were obtained overthe bottom half of the scattering plane with a small overlap into the upper half-plane to detail the on-axis forwardscattering pattern. The regions scanned in the measurement are also outlined in the upper panels of Figs. 3 and 4 forease of comparison with the numerical predictions. Note that although the measurements at ωa x / πc b = 0 .
077 and0.256 lie within two of the resonance bands identified in Fig. 1, no obvious additional resonant features are observedin the intensity maps that can be attributed to these resonance bands.The upper panels (a-c) of Figs. 3 and 4 show a prediction of the transmitted and reflected pressure intensitycalculated using two-dimensional multiple scattering theory (2D-MST). [4, 21] The MST accounts for elastic scatteringin the cylindrical tubes and assumes insonification by a cylindrical monopole source located at positions that match theexperiment. The experimentally measured acoustic intensities in panels (d-f) show good qualitative agreement withthe MST-predicted results. The reflected signal, R = 20 log [( P − P ) /P )], was estimated relative to the incidentamplitude P using the sound pressure level P measured in front of the incident face of the lens (regions to the left ofthe lens in Figs. 3 and 4). The measured reflection was at or below 13% of the incident amplitude ( R ≤ −
18 dB) overthe range of operational frequencies, which demonstrates significant transparency and is commensurate with recentFresnel lens designs in air. [23, 24] The measured reflection includes additional constructive interference compared tothe MST due to diffraction from the finite aperture size in the axial direction.In the forward direction a focusing peak is observed that strengthens in intensity and moves out away from the lenswith increasing frequency. The frequency-dependence of the focusing peak is expected. At very low frequencies theaperture is diffraction-limited; the wavelength is too long to adequately resolve the index gradient and there is alsosignificant diffraction around the aperture. As the frequency increases, the transmission approaches the geometriclimit of ray-acoustics where a focusing peak would be located beyond the focal point predicted by the ray tracing inFig. 2(a). At the lowest example frequency, ωa x / πc b = 0 .
077 [panels (a,d)], the lens is close to the diffraction limitwith height-to-wavelength ratio h/λ <
2. As the frequency is further reduced the focusing peak intensity becomessignificantly suppressed. Therefore, an operational bandwidth can be identified ranging between ωa x / πc b ≈ . x, y, z ) in front of an acoustic aperture located at x = 0 can be calculated by integrating over the aperture, [25] P ( x, y, z ) = xiλ Z Z A ˜ P ( y ′ , z ′ ) e ikr r dy ′ dz ′ (5)˜ P ( y ′ , z ′ ) = ψ ( y ′ , z ′ ) e iφ ( y ′ ,z ′ ) (6) r = x + ( y − y ′ ) + ( z − z ′ ) (7)where ψ ( y ′ , z ′ ) and φ ( y ′ , z ′ ) are the real-valued acoustic amplitude and phase of the complex pressure ˜ P ( y ′ , z ′ ) onthe aperture facet, and λ and k are the wavelength and wave number respectively. Here the integration aperture issimplified by the flat facet of our lens design, with the integration carried out over a simple rectangular area at the forward face of the lens.The pressure on the aperture facet ˜ P ( y ′ , z ′ ) can be calculated using anisotropic ray-tracing under the eikonalapproximation of Eq. (4). A schematic in Figure 5(a) depicts the ray trajectories of wave-fronts that emerge from thespherical source and traverse the forward facet of the lens. The calculation iterates over a discrete set of rays thatenter the rear facet (i.e. the incident face) at equally spaced intervals. The phase function φ ( y ′ , z ′ ) on the forwardfacet was determined by advancing the phase of each ray along the path lengths defined by ξ ( x ) assuming a constantphase at the source. The effective path length is altered by the spatially-dependent local sound speed in the lens.The phase difference ∆ φ = φ ( y ′ , z ′ ) − φ (0 ,
0) relative to the phase at the lens central axis is shown in Fig. 5(b), wherepositive or negative values represent a phase advance or delay respectively. The calculation is performed for a sourceposition on the lens central axis. Given the symmetry of the lens only one quadrant is shown. Curve fits in Fig. 5(d)indicate that ∆ φ is approximately quadratic in both the y - and z -directions.The amplitude function ψ ( y ′ , z ′ ) is inversely related to the square root of the local areal density of the rays. This wasapproximated by calculating the average nearest-neighbor separation of rays intersecting the forward facet; a relativeamplitude ¯ ψ ( y ′ , z ′ ) = ψ ( y ′ , z ′ ) /ψ (0 ,
0) can then be estimated from the inverse of this nearest-neighbor separation.Figure 5(c) shows the variation in ¯ ψ ( y ′ , z ′ ) after normalizing to the predicted amplitude in the absence of the lens.It is clear that the lens refraction has minimal impact on the amplitude ( < et. al. , [26] Eq. (5) is evaluated numerically within the 2D plane ofthe measurement using quadratic approximations to φ ( y ′ , z ′ ) and ψ ( y ′ , z ′ ). Intensity maps produced by the Rayleigh-Sommerfeld integration are shown in Fig. 3(g-i) and demonstrate significantly improved quantitative agreement withthe experiment. As was the case for the measurement and MST modeling, the Rayleigh-Sommerfeld-based pressureintensities are normalized to the pressure intensity of the spherical source in the absence of the lens. The 3D Rayleigh-Sommerfeld integration more accurately predicts both the location of the focal positions and their magnitude. Themeasured magnification in sound pressure level ranges between 4.5–7.3 dB over the operational bandwidth. The2D-MST predicts magnifications that are as much as 40% lower than the measured values, whereas the magnificationspredicted by the Rayleigh-Sommerfeld integration agree to within 10%. We emphasize that the 2D lattice only focusesalong one axis ( y -axis) resulting in lower magnifications than would be produced by axisymmetric 3D designs. [23, 24]The magnification can be increased by extending our 2D design to three dimensions using tubes bent into a toroidconfiguration. [12, 15]We now place our results in context with other metafluid design concepts. Traditional underwater and ultrasoundapplications have achieved an impedance-match by utilizing rubbers that have sound speeds less than water. Whilerubbers have been used as components in transparent metafluid lattices, [1] there have been other recent advances inacoustic impedance-matched designs. For example, negative-index complementary metamaterials have been proposedto significantly enhance the transparency through aberrating materials. [27] A class of metafluids based on the conceptof space-coiling have also featured prominently in the literature. [28–30] Space-coiled Fresnel lens designs have beenreported that show significant focusing while utilizing a thin aperture compared to the wavelength. [23, 24] Althoughthe strength of the space-coiling design is the ability to significantly modify phase within a confined space, there isa drawback that it can only delay the phase over a lengthened path; similar to rubbers in water, the effective soundspeed of these devices is less than the background fluid. The space-coiling design has not yet been demonstrated inwater, where a higher viscosity and lack of rigid boundary conditions may impose additional challenges.Our design extends the reach of aqueous transparent metafluids to include the option of phase advance and decreasedeffective path length. While not necessarily required for traditional lensing, there are metamaterial applicationsthat require higher sound speeds compared to the propagation medium, [17–20, 31] the most prominent of which isscattering reduction. The requirement of higher relative sound speeds features prominently in scattering reductiondesigns based on both coordinate transformation [32–34] and scattering cancellation. [35, 36] The realization of a high-sound speed, transparent GRIN lens represents an important proof-of-concept: it demonstrates that phase-advancemetafluids can be constructed with significant tunability in both sound speed and impedance, including the option ofmatched impedance. 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Am. , 471 (1949).[29] Z. Liang and J. Li, Phys. Rev. Lett. , 114301 (2012).[30] Y. Xie, A. Konneker, B.-I. Popa, and S. A. Cummer, Applied Physics Letters , 201906 (2013).[31] C. Garc´ıa-Meca, S. Carloni, C. Barcel´o, G. Jannes, J. S´anchez-Dehesa, and A. Mart´ınez, Phys. Rev. B , 024310 (2014).[32] S. A. Cummer and D. Schurig, New Journal of Physics , 45 (2007).[33] H. Chen and C. T. Chan, Applied Physics Letters , 183518 (2007).[34] S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr,Phys. Rev. Lett. , 024301 (2008).[35] M. D. Guild, M. R. Haberman, and A. Al`u, Phys. Rev. B , 104302 (2012).[36] T. P. Martin and G. J. Orris, Applied Physics Letters , 033506 (2012). a x a y xy (a) a y /a x P ha s e S peed ¯ c y y ¯ c xx (b) M X00.10.20.3 Γ ω a x / π c XM a y = a x (c) Y X00.10.20.3 Γ ω a x / π c XY a y = 2 a x (d) FIG. 1: (a) Schematic unit cell of the anisotropic lattice with a y = 2 a x . (b) Phase speeds in the homogenization limit of thelattice plotted as a function of anisotropy ratio a y /a x . Lines are fits to the data based on Ref. [21]. (c) Acoustic band structurecalculated using the Finite Element Method (FEM) for an isotropic lattice with tube outer radius R = 0 . a x . Blue lines plotthe acoustic sound speed in the ΓX direction. (d) Acoustic band structure calculated for an anisotropic lattice with tube outerradius R = 0 . a x . Blue and red lines plot the acoustic sound speed in the ΓX and ΓY directions respectively. Insets show thedirections of the bands in k -space. x (m) -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 y ( m ) -0.100.10.2 (a) x ( a x ) -4 -2 0 2 4 y ( a x ) ¯ c avg h/ (c) FIG. 2: (a) Ray paths through the lens (blue lines), which originate from a plane wave at normal incidence, are calculatedusing an eikonal approximation in the xy -plane; the red box indicates the location of the lens. Only rays incident on the upperhalf-plane of the lens are shown. (b) Schematic showing the locations of the lattice sites in the upper half-plane of the lens;the resulting sound speed profile is plotted to the right. The lattice sites are symmetric about y = 0. (c) Photograph of theassembled lens composed of 1 m long, hollow aluminum cylinders arranged in the desired lattice pattern. x (m) y ( m ) −0.5 0 0.5 1−0.4−0.20.20.40 x (m)−0.5 0 0.5 1 (d) (e) 0123x (m)−0.5 0 0.5 1(f) y ( m ) −0.4−0.20.20.40 y ( m ) Intensity at 7.272 kHz −0.4−0.20.20.40
Intensity at 13.052 kHz Intensity at 24.239 kHz (b)(a) (c) (g) (h) (i)
FIG. 3: Sound pressure intensity in the vicinity of the lens at ωa x / πc b = 0 .
077 (7.27 kHz), 0.138 (13.05 kHz), and 0.256(24.24 kHz) for a source location on the x -axis. (a-c) Pressure intensity calculated using 2D-MST. (d-f) Measured pressureintensity. (g-i) Pressure intensity calculated using a 3D Rayleigh-Sommerfeld approximation. Grey boxes show the position ofthe lens. Red outlines indicate the experimentally mapped areas. The reduced observational range of the reflected signal at24.24 kHz is due to a limited overlap between the incident and reflected signals in the time domain at high frequency. Intensity at 13.052 kHz x (m) y ( m ) −0.5 0 0.5 1 −0.4−0.20.20.40 x (m)−0.5 0 0.5 1 x (m)−0.5 0 0.5 1 0123 (d) (f) y ( m ) Intensity at 7.272 kHz −0.4−0.20.20.40
Intensity at 24.239 kHz (b)(a) (c) (e)(e)
FIG. 4: Sound pressure intensity in the vicinity of the lens at ωa x / πc b = 0 .
077 (7.27 kHz), 0.138 (13.05 kHz), and 0.256(24.24 kHz) for a source positioned at a 15 ◦ angle with respect to the x -axis. (a-c) Pressure intensity calculated using 2D-MST.(d-f) Measured pressure intensity. Grey boxes show the position of the lens. Red outlines indicate the experimentally mappedareas. (b) ∆ φ ( y ′ , z ′ ) y ′ (m) z ′ ( m ) -120 -80 -40 0 40 (c) ¯ ψ ( y ′ , z ′ ) y ′ (m) Position (m) ∆ φ ( D e g r ee s ) -150-100-50050 (d) y ′ -axis ( z ′ = 0) z ′ -axis ( y ′ = 0) FIG. 5: (a) Schematic showing the experimental setup; blue lines indicate ray paths emerging from the spherical source andRayleigh-Sommerfeld integration vectors from the forward facet of the lens to hydrophone locations in the measurement plane.(b) Change in phase ∆ φ ( y ′ , z ′ ) (degrees) over the forward facet (due to symmetry only the 1st quadrant is shown). Blue regionsrepresent a phase delay, red regions represent a phase advance. (c) Normalized amplitude ¯ ψ ( y ′ , z ′ ) over the forward facet, wherenormalization is with respect to the amplitude of the spherical wave in the absence of the lens. (d) Red and blue lines showthe phase change ∆ φ plotted as a function of position along the y ′ - and z ′′