Focusing Ultrasound with Acoustic Metamaterial Network
Shu Zhang, Leilei Yin and Nicholas Fang
Department of Mechanical Science and Engineering, University of Illinois at Urbana- Champaign, USA
We present the first experimental demonstration of focusing ultrasound waves through a flat acoustic metamaterial
lens composed of a planar network of subwavelength Helmholtz resonators. We observed a tight focus of half-
wavelength in width at 60.5 KHz by imaging a point source. This result is in excellent agreement with the numerical
simulation by transmission line model in which we derived the effective mass density and compressibility. This
metamaterial lens also displays variable focal length at different frequencies. Our experiment shows the promise of
designing compact and light-weight ultrasound imaging elements.
PACS numbers: 43.20. +g,43.35.+d,46.40.Ff
High-resolution acoustic imaging techniques are the
essential tools for nondestructive testing and medical screening.
However, the spatial resolution of the conventional acoustic
imaging methods is restricted by the incident wavelength of
ultrasound. This is due to the quickly fading evanescent fields
which carry the subwavelength features of objects. To
overcome this diffraction limit, a remarkable perfect lens is
proposed by John Pendry , which offers the promise to build
a device allowing super-resolution imaging of an object. This
perfect lens is based on focusing the propagating wave and
recovering the evanescent field through a flat negative-index
slab. Since then research on metamaterials has been stimulated
by the opportunity to develop artificial media that refracts
waves in negative direction. Several different metamaterials
have been proposed and demonstrated to present negative index
of refraction [2-6].
The successful demonstration of electromagnetic (EM)
superlens [7-10] has inspired the search for the analogous
acoustic negative-index lens. In fact, phononic crystals [11-15]
were first investigated to develop negative-refractive devices
for sound waves. Beam steering in phononic crystals can be
achieved by Bragg scattering, leading to enhanced diffraction
in negative direction. Ultrasound focusing from negative
refraction by a three-dimensional phononic crystal was first
demonstrated experimentally by Yang et al..A focal spot
around five wavelengths in width was observed in the far field
at 1.57 MHz. Recently, a finer resolution was achieved by
focusing the ultrasound field emitted by a subwavelength line
source using a two-dimensional (2D) phononic crystal
However, for lens design based on phononic crystals, the
dependence of band structure on the lattice periodicity usually
requires the spatial modulation to be the same order of
magnitude as the acoustic wavelength, which would makes
such structure impracticably large. Locally resonant sonic
materials  made a major step towards the acoustic
metamaterial development. Since the lattice constant is much
smaller than the relevant wavelength, effective medium
properties can be attributed to this sonic material at low
frequency. With appropriate resonances included into the
building block, acoustic metamaterials with either negative
effective mass density or bulk modulus or both have been
demonstrated [17-20]. These anomalous phenomena resulted
from strong coupling of the traveling elastic wave in the host
medium with the localized resonance in the building block.
However, to the best of our knowledge, there is no
experimental demonstration of focusing ultrasound waves in
these negative index acoustic metamaterials.
In this paper, we experimentally investigated the focusing
of a point source from a designed ultrasonic metamaterial
consisting of a planar network of subwavelength Helmholtz
resonators. To facilitate the design, we adapted the 2D
transmission line (TL) method which is widely used in the
development of negative index EM metamaterials [8-10].In this
approach, the acoustic system is converted to an analogous
lumped circuit model in which the motion of the fluid is
equivalent to the behavior of the current in the circuit. Similar
to permittivity and permeability in the EM metamaterial , the
effective density and compressibility of the network structure
are found to be related to the capacitance and inductance in this
lumped circuit. Earlier, in the one-dimensional version of this
ultrasonic metamaterial, the elastic modulus is found to be
negative at specific frequency range theoretically and
Figure 1 shows the experimental setup to study the
focusing phenomena of the acoustic metamaterial. To prepare
the sample, we machined a 2D array of periodically connected
subwavelength Helmholtz resonators in an aluminum plate and
the resonators are filled with water. As shown in previous work
[22-24], a main transmission channel with recurrent side
branches, which are closed at the outer end, is analogous to a
circuit of a series of inductors with shunt capacitors. On the
other hand, when the side tubes inserted in the main channel is
open on the outer end, the acoustic system can be described by
a lumped network of a series of capacitors with shunt
inductors. The left and right half parts in the sample are 2D
periodic versions of those different types of topology
respectively. One unit cell from each half part is enlarged and
shown in the two insets respectively.
FIG. 1 (color). Schematic showing the experimental setup. The
sample with PI/NI interface is composed of an array of
different designed Helmholtz resonators machined from an
aluminum plate. Unit cells of each half part and the
corresponding inductor–capacitor circuit analogy are shown in
The left half part is composed of a 2D array (40×40) of
larger cavities connected with main channels. The volume of
the cavity is around ten times of that of one section of the
channels. Consequently, when an incident acoustic wave is
applied onto the fluid in the channels, the pressure gradient
through the channels is much greater than that inside the cavity.
Hence, it is as if the fluid in the cavity were at rest relative to
those in the channels .So when the plug of fluid in the
channels oscillates as a unit, there are adiabatic compressions
and rarefactions of the fluid inside the larger cavity. Such an
acoustic system is analogous to an inductor–capacitor circuit as
shown in the inset with the channels acting as a series of
. The periodicity (3.175mm) of the sample is
one-eighth of the wavelength at around 60 KHz frequency
range. Given this value, the lumped circuit model is a valid
approximation for the distributed acoustic system with only
10% error . Following the approach of EM circuit
analysis[8-10], the effective density and compressibility of this
L and the cavity providing the stiffness element as
network can be expressed in the form as
d is the periodicity,
S is the cross
section area of the channels. Both effective density and
compressibility are positive. Effective relative acoustic
P n can be determined by
w L C
effective positive index (PI) medium.
The right half part of the sample is the dual configuration
of the left half part, in which there is an array (40×40) of
orifices connected with channels. The volume of one section of
the main channel is designed as around ten times of that of the
orifice. Since the fluid in the orifice is not confined, it
experiences negligible compression while the fluid in the
channels experience less movement in average compared with
that in the orifice . Consequently, when the fluid in the
orifice oscillates as a unit, there are adiabatic compressions and
rarefactions of the fluid inside the main channels. Such an
acoustic system is described as a lumped network with a series
for the main channel part and a shunt
due to the orifice. The periodicity is the same
as that in the left part, so the effective mass density and
compressibility can be
L d S
cross section area of connecting channels. Both parameters are
negative. The refractive index
c is speed of sound in water. We call this half part as
similarly estimated as
field map of the acoustic NI metamaterial and (c) Line plot of
pressure field cross the focal plane parallel to interface
For numerical verification, lumped circuit simulation of
this acoustic network was performed by using commercial
circuit simulator SPICE. Comparison of Fig. 2 (a) and (b)
shows that the field plots found through simulation is in
remarkable agreement with the experimental results. In Fig.
2(c), the measured data in blue line is shifted to left by 3.175
mm for comparison purpose. The comparison demonstrates a
very good match in the focus width between the measurement
and the numerical simulation. We also plotted the full width at
half maximum (FWHM) at different frequencies in Fig. 3(a).
d is periodicity and
S is the
= = −
negative. So this network structure acts as a medium exhibiting
negative index (NI) of refraction. The two half parts are
designed with effective indices of equal and opposite value and
at the design frequency 60.5
For experimental confirmation of ultrasound focusing in
this acoustic metamaterial, we measured the pressure field
through this PI/NI interface. The ultrasound waves were
launched from a horn shaped transducer with a tip of 3mm
diameter in size. The tip is inserted into a hole drilled through
the center of the PI part ((column, row) = (20, 20)) to
illuminate the sample. To map the pressure field, a hydrophone
was mounted on two orthogonal linear translation stages. By
stepping the hydrophone to the positions above those through
holes in the NI part and recording the pressure amplitude at
every step, we acquired the spatial field distribution of the
ultrasound wave focusing pattern.
Fig.2 (a) shows the pressure field map in the NI part at
60.5 KHz with the PI/NI interface along x=0. The pressure
amplitude is normalized to unity. A tight spot is observed in
experiment as is evident from the plot. The pressure cross the
focal plane along y direction is plotted in Fig. 2(c). The full
width at half maximum (FWHM) was found to be 12.2mm,
corresponding to a resolution of 0.5 wavelength in water.
FIG. 2 (color). Pseudo colormap of the normalized pressure
field distribution at 60.5 KHz. (a) Measured and (b) simulated
The optimal focus imaging is observed at 60.5 KHz from both
experimental and numerical results.
The focal length defined as the distance between focus and
PI/NI interface is plotted in Fig.3 (b) as a function of
frequency. Ray acoustics is utilized to estimate the focal length
as shown in the blue solid line. The magnitude of the refractive
index in the acoustic metamaterial decreases from 1.19 to 0.85
as the frequency increases from 56 to 66 KHz. And this
analysis predicts that the negative refractive index approaches -
1 relative to the PI part at 60.5 KHz. The decrease of the
index magnitude over this frequency range causes the focal
length decreasing from 79.27 to 37.6mm. The lumped circuit
simulation gives the dashed green curve while the red stars
show the measurement data. The three curves present a good
match in trend. However, around 10 mm shift is observed and
we are investigating this discrepancy.
FIG. 3 (color). (a) Measured and calculated FWHM of the
focus as a function of frequency. Blue solid line is calculated
using acoustic circuit model and red circles represent
experimental data. (b) Measured and calculated focal length as
a function of frequency. The analytical curve (blue solid line) is
based on ray propagation and the green dashed line is
numerical results from acoustic circuit model. Experimentally
obtained data are shown by red circles.
In order to achieve high-quality focus imaging, the ratio
of refractive index should be -1 at the PI/NI interface. Only
when the index is matched, based on ray acoustics, the angle
of refraction equals the angle of incidence for each ray such
that all rays can be brought to the same focal spot in the NI
part. However, due to the loss and variation of the inductance
and capacitance from their design values resulted from
machining tolerance; the refractive index is not exactly
matched in the measurement. Therefore, the distance between
the focus and the interface varies for different incident angle
as result of aberration . Due to this index mismatch, we
observed that the focal spot elongated along x direction while
remain narrow along the direction parallel to the interface in
the experiment. And the focus is in a position closer to the
interface than the source. The best focusing resolution is
observed at 60.5 KHz. We expect that the ratio of refractive
index might approach -1 at lower frequency. However
verification of this is beyond the operation frequency range of
our transducer in the experiment. The slight material loss in
the measurement also significantly degrades the focusing
resolution as shown in several papers [26-27]. It was noted
that single PI/NI interface does not allow the enough growth
of evanescent fields to achieve sub diffraction focusing 
while sandwich structure (two PI/NI interfaces) offers better
chance to overcome the diffraction limit .
In summary, the emission of a point source at kilohertz
frequency was brought to a focus through the PI/NI interface
because of the negative refraction in this ultrasonic
metamaterial, which is expected to be a step toward a novel
acoustic imaging lens. The resolution of 0.5wavelength was
recorded by focusing the acoustic field of a point source. This
is not sub diffraction imaging, but among the best achievable
passive acoustic imaging elements. The unit cell of the acoustic
network is only one eighth of the operating wavelength,
making the lens in a compact size. Compared with
conventional lenses, the flat thin slab lens takes advantages in
that there is no need to manufacture the shapes of spherical
curvatures and the focus position is insensitive to the offset of
source along the axis. Also this negative index lens offers
tunable focal length at different frequencies. More generally,
this design approach may lead to novel strategies of acoustic
cloak for camouflage under sonar.
This work is partially supported by DARPA grant HR0011-05-
3-0002. We are grateful to the inspiring private communication
with Prof. Jianyi Xu  from Nanjing University, China at the
initial stage of the sample design and the helpful discussion
with Prof. William O’Brien at UIUC regarding experimental
 J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
 D. R. Smith et al, Phys. Rev. Lett, 84, 4184 (2000).
 R. A. Shelby, D. R. Smith and S. Schultz, Science 292, 77
 S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis,
Phys. Rev. Lett, 90, 107402 (2003).
 A. A. Houck, J. B. Brock and I. L. Chuang, Phys. Rev.
Lett. 90, 137401 (2003).
 H. J. Lezec, J. A. Dionne, H. A. Atwater, Science 20,430
 N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534
 G. V. Eleftheriades, A. K. Iyer, P. C. Kremer, IEEE Trans.
Microwave Theory and Tech. 50, 2702 (2002).
 A. K. Iyer, P. C. Kremer, G. V. Eleftheriades, Opt.
Express 11, 696 (2003).
 A. Grbic, G. V. Eleftheriades’, Phys. Rev. Lett. 92,
 C.Y. Qiu, X. D. Zhang and Z. Y. Liu, Phys. Rev. B 71,
 X. D. Zhang and Z. Y. Liu, Appl. Phys. Lett. 85, 341
5 Download full-text
 S. X. Yang et al., Phys. Rev. Lett. 93, 024301( 2004).
 M. Z. Ke et al., Phys. Rev. B 72, 064306 (2005).
 A. Sukhovich, L. Jing and J. H. Page, Phys. Rev. B 77,
 Z. Liu, X. Zhang, Y. Mao et al., Science 289, 1734 (2000).
 J. Li and C. T. Chan, Phys. Rev. E 70, 055602 (2004).
 Z. Y. Liu, C. T. Chan, and P. Sheng, Phys. Rev. B 71,
 Y. Q. Ding, Z.Y. Liu ,C. Y. Qiu and J. Shi , Phys. Rev. Lett.
99, 093904 (2007).
 J. Li, Z. Liu, and C. Qiu, Phys. Rev. B 73, 054302 (2006).
 N. Fang, D. J. Xi, J. Y. Xu, M. Ambrati, W. Sprituravanich,
C. Sun, and X. Zhang, Nat. Mater. 5, 452 (2006).
 L. E. Kinsler, Fundamentals of Acoustics, 1982, Wiley,
 G. W. Stewart, “Acoustic wave filters”, Phys.Rev.20,528
 W. P. Mason, Bell Syst. Tech. J . 6, 258 (1927).
 L.L.Beranek, Acoustics,1954,McGRAW-HILL,New York.
 D. R. Smith et al, Appl. Phys. Lett. 82 ,1506 (2003).
 N. Fang and X. Zhang, Appl. Phys. Lett. 82, 161 (2003).
 J.Y. Xu, private communication.