Design and demonstration of broadband thin planar diffractive acousticlenses

Wenqi Wang, Yangbo Xie, Adam Konneker, Bogdan-Ioan Popa, and Steven A. Cummera)

Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA

(Received 21 July 2014; accepted 1 September 2014; published online 11 September 2014)

We present here two diffractive acoustic lenses with subwavelength thickness, planar profile, and

broad operation bandwidth. Tapered labyrinthine unit cells with their inherently broadband

effective material properties are exploited in our design. Both the measured and the simulated

results are showcased to demonstrate the lensing effect over more than 40% of the central

frequency. The focusing of a propagating Gaussian modulated sinusoidal pulse is also

demonstrated. This work paves the way for designing diffractive acoustic lenses and more

generalized phase engineering diffractive elements with labyrinthine acoustic metamaterials.VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895619]

Acoustic lenses manipulate wave propagation and con-

trol energy transfer through modulating the phase and ampli-

tude of the incident wave. They are widely applied in

acoustic imaging,1 therapeutic ultrasound,2 and sonar sys-

tems.3 Traditional acoustic lenses are often constructed with

homogeneous materials chosen from a limited material

library and face challenges in increasing throughput and

reducing device footprints. During the past few years, the

emergence of acoustic metamaterials,4–12 together with the

related design tools, such as transformation acoustics13,14

and computational optimization algorithms, provide alterna-

tive solutions to acoustic lenses designer.

Diffractive lenses operate by generating interference

and diffraction patterns through phase engineering. Their

mechanism is different from gradient-index (GRIN)

lenses,15,16 which rely on the gradually varying refractive

index to accumulate the phase delay needed. Diffractive

lenses thus are usually advantageous with their thinness,

high transmission, and larger degree of design freedom.

However, achieving a broad range of phase modulation

while retaining high transmission puts demanding constraints

on the wave modulating elements. Another challenge lies in

the operating bandwidths as the properties of diffractive ele-

ments are usually sensitive to frequency, which can limit the

lensing effect to a narrow frequency range.

With both challenges in mind, we chose tapered labyrin-

thine acoustic metamaterials17,18 for designing broadband

diffractive lenses. The space-coiling geometry of this type of

unit cells renders them inherently broadband for lenses. The

effective refractive index approximates the ratio between the

inner meandering path and the unit cell diameter and thus

the phase change in the unit cell is proportional to the operat-

ing frequency, which will result in a broadband diffractive

device. Here, we adopt a frequency normalized phase modu-

lation D/ðxÞ x0

x in the design so that the broadband perform-

ance of both the unit cells and the lenses can be quantified

with this parameter. Figure 1(a) demonstrates the perform-

ance of a typical tapered labyrinthine unit cell. Its frequency

normalized phase modulation varies with only a small slope

over a frequency span of 1400 Hz around the central fre-

quency of 3000 Hz and the amplitude of its transmission

coefficient jS21j remains a relatively constant value around

70%.

Two lenses are demonstrated here. The first lens can

transform cylindrical radiation to a plane wave, or vice versa

(referred to later as point-to-plane lens, whose ray trajectory is

shown in Figure 1(b)). The second lens can transform the radi-

ation from a point source to an image spot (referred to later as

point-to-point lens, whose ray trajectory is also shown in

Figure 1(b)). Both lenses have the same thickness, which

ranges approximately from 0.5 to 0.75 wavelengths across the

frequency range of interest (2400 Hz–3600 Hz). Simulated

and measured results are compared, demonstrating that both

FIG. 1. Broadband characteristics of a typical unit cell and ray trajectory sche-

matic diagrams and phase modulation profile for both lenses. (a) A tapered lab-

yrinthine unit cell and the dispersion curves of its transmission amplitude and

normalized phase modulation. (b) Ray trajectory schematic for both lenses and

the normalized phase modulation profile (unit: rad/(2p Hz)).a)Electronic mail: cummer@ee.duke.edu

0003-6951/2014/105(10)/101904/3/$30.00 VC 2014 AIP Publishing LLC105, 101904-1

APPLIED PHYSICS LETTERS 105, 101904 (2014)

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lenses proposed exhibit excellent lensing effect as desired,

high energy throughput, and bandwidth of more than 10% of

the central frequencies.

Both lenses presented here are composed of a series of

carefully designed and optimized labyrinthine unit cells.

Labyrinthine acoustic metamaterials17–20 acquire their prop-

erties mostly from their inner meandering geometries, thus

avoiding the drawbacks of narrow bandwidth and high

absorption which are typical of most of the locally resonant

metamaterials (such as those based on Helmholtz resonators

or elastic membranes). In the lens design process, analytical

estimates are formulated to predict the phase modulation

needed for each type of the lens. Then finite element analysis

(FEA) is used for designing the unit cells suited for the

lenses and simulating the lensing performance. Experimental

prototypes were fabricated using fused filament fabrication

(FFF) 3D printing and the field mapping measurements are

performed in our lab-made two-dimensional parallel plate

acoustic scanning measurement system.16,21

A point-to-plane lens capable of converting a cylindrical

radiation to a plane wave is required to possess a frequency

normalized phase modulation profile (the phase change

between the exiting and the entrance interfaces of the lens)

along the interface

D/ xð Þx0

x¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2

� �2 þ f 2

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ f 2

pc

x0; (1)

where f is the focal length, L is the width of the lens, x is the

distance from the center to a point along the interface, c is the

speed of sound in air. A lens composed of 48 labyrinthine unit

cells approximating the phase modulation profile given by Eq.

(1) was designed and fabricated. Figure 2(a) shows the simu-

lation results at 3000 Hz, where the conversion from cylindri-

cal radiation to a plane wave is clearly seen. Figures 2(b)–2(d)

shows the measurement results at 2800 Hz, 3000 Hz, and

3200 Hz, all of which agree well with the simulation. The field

patterns at these three different frequencies indicate that the

lens can operate over a bandwidth of more than 400 Hz, which

is over 10% of the 3000 Hz central frequency. Such opera-

tional bandwidth is broader than most of the resonant

metamaterial-based modulation devices.

FIG. 2. Point-to-plane lens converting a cylindrical radiation to a plane

wave. (a) Simulation result at 3000 Hz. The red rectangular region indicates

the measurement area in experiment. (b) Measured field patterns at 3000 Hz.

(c) Measured field patterns at 2800 Hz. (d) Measured field patterns at

3200 Hz.

FIG. 3. Point-to-point lens projecting

cylindrical radiation to a focal spot. (a)

Simulated pressure field pattern (real

value) at 3000 Hz. (b) Simulated pres-

sure amplitude pattern at 3000 Hz. (c)

Measured field patterns at 2800 Hz,

3000 Hz, and 3200 Hz (left: amplitude

distribution, right: real value of the

pressure field). (d) Measured amplitude

profiles along y¼ 0.6 m at 2800 Hz,

3000 Hz, and 3200 Hz, compared with

that in the free space.

101904-2 Wang et al. Appl. Phys. Lett. 105, 101904 (2014)

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A point-to-point lens capable of transforming a cylindrical radiation to a focal spot must have the following phase modula-

tion profile:

D/ xð Þx0

x¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL

2

� �2

þ f 21

sþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL

2

� �2

þ f 22

s

c�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ f 2

1

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ f 2

2

pc

0BB@

1CCA

x0; (2)

where f1 is the length from the point source to the lens, f2 is

the length from the lens to the focal spot, and other variables

have the same meaning as defined above. The lens is com-

posed of 48 labyrinthine unit cells with phase modulation

approximating the lens design Eq. (2).

The simulated field patterns are plotted in Figures 3(a)

and 3(b), where a cigar-shaped focal spot can be clearly

identified. The measured field patterns are plotted in Figure

3(c), which match the simulated results well. Comparing the

field patterns at 2800 Hz, 3000 Hz, and 3200 Hz, we can con-

clude that the operating bandwidth of the point-to-point lens

is more than 400 Hz (note that we expect 1400 Hz bandwidth

from the cell design). Figure 3(d) plots the measured ampli-

tude along y¼ 0.6 m for each of these three frequencies,

whose energy density at the focal point is magnified by a fac-

tor of approximately 15 compared to that of the free space

propagation.

The time evolution of a broadband pulse after being

modulated by the point-to-point lens is also studied here.

A Gaussian modulated sinusoidal pulse with a central

frequency of 3000 Hz and a full-width-at-half-maximum

(FWHM) bandwidth of 1365 Hz was chosen as the input.

The field pattern as a function of time was measured and

three representative frames are shown in Figure 4. A frame

at the initial stage of the formation of the focused pulse after

the modulation of the lens is first plotted in Figure 4(a), and

its time is defined as a reference t0. Two other frames at

t0þ 566.9 ls and t0þ 1133.8 ls are shown, respectively, in

Figures 4(b) and 4(c). The pulse remained compact along its

propagation along the approximately 40 cm distance in free

space. The generation of such high power density acoustic

pulse, similar to the recently proposed “sound bullet,”22

might be useful for various non-destructive therapeutic

treatments.

The two lenses demonstrated here serve as examples of

possible wavefront shaping devices that can be designed

with the phase modulating labyrinthine metamaterials. More

generalized holographic diffractive elements23 adopt similar

phase engineering and can be designed in similar

approaches. To further improve the performance of the

lenses presented here, optimization algorithms such as

genetic algorithm24 can be utilized for the purpose of reduc-

ing undesired scattering, increasing throughput, etc.

In conclusion, we present in this paper the design and

tests of two broadband subwavelength-thin planar diffractive

acoustic lenses, both of which are constructed with tapered

labyrinthine unit cells. The measured lensing effects are in

excellent agreement with the simulations, and broadband

measurements show an operating bandwidth of more than

40% of the central frequency. Our work provides a labyrin-

thine metamaterial based design approach for low profile

acoustic lenses and might be useful for applications such as

therapeutic ultrasound, acoustic imaging, and sonar systems.

This work was supported by a Multidisciplinary

University Research Initiative from the Office of Naval

Research (Grant No. N00014-13-1-0631).

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FIG. 4. Measured focused Gaussian modulated sinusoidal pulse propagating

in free space. Three frames at (a) t0, (b) t0þ 566.9 ls, and (c) t0þ 1133.8 ls

show the evolution of the pulse propagating in free space.

101904-3 Wang et al. Appl. Phys. Lett. 105, 101904 (2014)

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