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: [email protected]
0003-6951/2014/105(10)/101904/3/$30.00 VC 2014 AIP Publishing LLC105, 101904-1
APPLIED PHYSICS LETTERS 105, 101904 (2014)
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.3.216.26
On: Thu, 18 Sep 2014 16:14:34
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)
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.3.216.26
On: Thu, 18 Sep 2014 16:14:34
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).
1W. S. Gan, Acoustical Imaging: Techniques and Applications forEngineers (John Wiley & Sons, New York, 2012).
2H. Azhari, Basics of Biomedical Ultrasound for Engineers (John Wiley &
Sons, New York, 2010).3E. O. Belcher, D. C. Lynn, H. Q. Dinh, and T. J. Laughlin, in OCEANS’99MTS/IEEE. Riding the Crest into the 21st Century (IEEE, 1999), Vol. 3,
p. 1495.4Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C. Chan, and P. Sheng,
Science 289, 1734 (2000).5J. Li and C. T. Chan, Phys. Rev. E 70, 055602 (2004).6N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X.
Zhang, Nat. Mater. 5, 452 (2006).7J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, Nat. Mater. 8, 931 (2009).8A. N. Norris, J. Acoust. Soc. Am. 125, 839 (2009).9S. Zhang, L. Yin, and N. Fang, Phys. Rev. Lett. 102, 194301 (2009).
10D. Torrent and J. S�anchez-Dehesa, Phys. Rev. Lett. 105, 174301 (2010).11A. C. Hladky-Hennion, J. O. Vasseur, G. Haw, C. Cro�enne, L. Haumesser,
and A. N. Norris, Appl. Phys. Lett. 102, 144103 (2013).12R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Al�u,
Science 343, 516 (2014).13S. A. Cummer, M. Rahm, and D. Schurig, New J. Phys. 10, 115025
(2008).14H. Chen and C. T. Chan J. Phys. D: Appl. Phys. 43, 113001 (2010).15T. P. Martin, M. Nicholas, G. J. Orris, L. W. Cai, D. Torrent, and J.
S�anchez-Dehesa, Appl. Phys. Lett. 97, 113503 (2010).16L. Zigoneanu, B.-I. Popa, and S. A. Cummer, Phys. Rev. B 84, 024305
(2011).17Y. Xie, A. Konneker, B.-I. Popa, and S. A. Cummer, Appl. Phys. Lett.
103, 201906 (2013).18Z. Liang and J. Li, Phys. Rev. Lett. 108, 114301 (2012).19Y. Xie, B.-I. Popa, L. Zigoneanu, and S. A. Cummer, Phys. Rev. Lett. 110,
175501 (2013).20Z. Liang, T. Feng, S. Lok, F. Liu, K. B. Ng, C. H. Chan, J. Wang, S. Han,
S. Lee, and J. Li, Sci. Rep. 3, 1614 (2013).21B.-I. Popa, L. Zigoneanu, and S. A. Cummer, Phys. Rev. Lett. 106,
253901 (2011).22A. Spadoni and C. Daraio, Proc. Natl. Acad. Sci. 107, 7230 (2010).23J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York,
1998).24D. Li, L. Zigoneanu, B.-I. Popa, and S. A. Cummer, J. Acoust. Soc. Am.
132, 2823 (2012).
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)
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.3.216.26
On: Thu, 18 Sep 2014 16:14:34