Ultrasonics 42 (2004) 787–790

www.elsevier.com/locate/ultras

Ultrasonic band gaps and negative refraction

M. Torres a,*, F.R. Montero de Espinosa b

a Instituto de F�ısica Aplicada, Consejo Superior de Investigaciones Cient�ıficas (CSIC), Serrano 144, 28006 Madrid, Spainb Instituto de Ac�ustica, Consejo Superior de Investigaciones Cient�ıficas (CSIC), Serrano 144, 28006 Madrid, Spain

Abstract

The first experimentally observed ultrasonic band gaps in periodic bidimensional composited are reviewed here. The studied basic

structure consists of an aluminum alloy plate with a periodic arrangement of cylindrical holes filled with mercury. Localization

phenomena in linear and point defects have been observed and an ultrasonic waveguide capable to bend the ultrasonic radiation by

90� has been achieved. We also revisit twinned-square periodic structures for ultrasonic wave bending and splitting. Such devices

allow 45� bending of waves, whereas an extreme anomalous refraction law at the grain boundaries has also been experimentally

observed. Finally, a preliminary study about the possibility of ‘‘left-handed’’ behaviour in ultrasonic crystals is presented in this

work. The device consists of a slab of the above mentioned metallic composite attached to an epoxy wedge. In this system, clues to

negative refraction are theoretically shown.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Negative refraction; Ultrasonic left-handed materials; Ultrasonic band gaps

Interest in wave propagation and scattering in inho-

mogeneous periodic media has been stirred after the

discovery of photonic crystals [1,2], which motivated the

search for analogous phenomena in elastic [3–7] and

sonic [8,9] band gap materials. In these materials,interesting effects of wave localization [10], wave bend-

ing [10–12], tunneling [13], focusing [14] and imaging

with a new sonic plane lens [15] appear. Furthermore, an

extreme anomalous ultrasonic refraction law has also

been experimentally observed at the boundaries of

twinned periodic structures [12]. After revisiting some of

these recent experiments, we shall present here a wave

simulation to show that ultrasonic negative refraction isalso theoretically possible, as it has been recently shown

in photonic crystals [16].

The studied basic structure consists of an aluminum

alloy plate with a periodic square arrangement of

cylindrical holes filled with mercury and the frequency

range of the experiments and theoretical calculations is

from 0.3 to 1.5 MHz. The acoustic attenuation of the

samples was measured using the gain-phase module ofa HP 4194 A impedance gain-phase analyzer, as de-

*Corresponding author. Tel.: +34-91-561-8806; fax: +34-91-411-

7651.

E-mail address: manolo@iec.csic.es (M. Torres).

0041-624X/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.ultras.2004.01.041

scribed elsewhere [5,6]. The finite-difference time-do-

main (FDTD) method was used to fit the mentioned

attenuation experiments [6]. Typical measurements of

attenuation along the two main crystallographic direc-

tions, for a filling ratio f of 0.4, are shown in Fig. 1. Anarrow full band gap is observed between 0.77 and 0.8

MHz [12].

To study localized states, we have substituted one

mercury cylinder by a piezoelectric vibrator that we have

excited at a frequency within the band gap, and at a

frequency clearly outside the gap [10]. Scans of the

vibration amplitude at the surface sample were per-

formed as shown in Fig. 2. In the first case, Fig. 2(a), thelocalization of the wave strongly appears. The wave

localization at frequencies in the center of the full gap is

isotropic. As the frequency increases to the edge of the

gap, the exponential decaying length increases linearly.

At a wave frequency far away from the gaps, as it is

expected, the wave is completely delocalized and spreads

out, Fig. 2(b).

As an example of linear defect, in Fig. 3 we show thepropagation of ultrasound when two rows of mercury

cylinders are missing according to an inverted L-shaped

structure [10]. A monochromatic excitation signal cor-

responding to the center of the band gap along the

(1 0 0) main symmetry direction, was launched along the

Fig. 1. Typical attenuation curves for the aluminum–mercury periodic

composite with a filling ratio f of 0.4. (a) Along (1 0 0) direction. (b)

Along (1 1 0) direction. The sample is made of aluminum alloy (AL-

PLAN MEC 7079 T 651) of 2 cm thickness. The plate is drilled with

holes of 2 mm diameter forming a square periodic structure with lattice

parameter of 2.8 mm. A narrow full band gap is observed between 0.77

and 0.8 MHz.

Fig. 2. (a) Localization of an ultrasonic wave in the aluminum–mer-

cury periodic composite. A 3D contour plot of the surface amplitude

scan of the periodic composite with a point defect is shown. A cylinder

has been substituted by a piezoelectric vibrator excited at a frequency

within the band gap. An array with 36 holes is scanned. (b) When the

piezoelectric column excites the sample at a frequency of 1.3 MHz,

clearly out of the gap, the wave is delocalized and spreads out.

Fig. 3. Surface amplitude scan of a periodic metallic composite with an

inverted L-shaped linear defect showing the wave-guiding. An array

with 165 network points is scanned. In the indicated region marked

with superimposed lines, two rows of mercury cylinders are missing.

Note that the radiation is guided along the path with no holes.

788 M. Torres, F.R. Montero de Espinosa / Ultrasonics 42 (2004) 787–790

missed rows scanning the vibration amplitude at the

surface of the sample. In this experiment, the ultrasonic

radiation is guided along the path free of holes and is

able to illuminate the right arm of the inverted L-shapedpath. By using the FDTD method applied to the same

aluminum–mercury composite, this wave-guiding effect

has also been theoretically shown [11].

A twinned device consisting of four ultrasonic

monocrystalline domains arranged in the two main

crystallographic directions, in such a way that pairs of

same oriented domains are opposite and both orienta-

tions are interwoven, was recently presented [12] (seeFig. 4). This structure is able to bend and split ultrasonic

waves and does not require the existence of a complete

band gap. By using the FDTD commercial program

Wave 2000, a cylindrical ultrasonic wave is generated by

a point source at a frequency of 0.875 MHz and its

propagation from pure aluminum to the aluminum–

mercury composite has been calculated and is shown in

Fig. 5. This pattern resembles that shown in Fig. 13(b)of the Ref. [16] which has been interpreted as an exotic

focusing phenomenon and a signature of negative re-

fraction in photonic crystals.

The FDTD program Wave 2000 allows us to simu-

late the experiments with the same time evolution as

they are carried out. As it is clearly observed in Fig. 5,

the field of wave velocities is strongly anisotropic in the

ultrasonic crystal. The easy propagation direction

coincides with the main crystallographic (1 0 0) direc-tion. The wave velocity along the (1 0 0) direction is

p2

Fig. 4. Ultrasonic twinned crystal device. A, A0 and B, B0, denote the

crystalline domains oriented along the (1 0 0) and (1 1 0) directions,

respectively.

Fig. 5. Theoretical calculation showing the anisotropy of the ultrasonic

wave propagation within the periodic metallic composite. A cylindrical

ultrasonic wave propagates from pure aluminum (right) to the alu-

minum–mercury ultrasonic crystal (left). The vertical line indicates the

boundary between both regions. The mercury cylinders are indicated

by superimposed circles. The wave fully couples to the ultrasonic

crystal lattice. Starting from this pattern, the square shape of the

corresponding isofrequency curve can be drawn, as it is shown in Fig. 7

(left and top-right).

Fig. 6. Surface scanning of the vibration amplitude for a 6 · 7 square

array of mercury cylinders. The unit cell is shown by small circles at the

top-left corner. The central hole of the array, marked with x, is

substituted by a piezoelectric vibrator impulsively excited. The wave

couples to the square lattice and becomes a square wave. The wave-

crystal coupling is already fully established at the first neighbors of the

piezoelectric position.

Fig. 7. Scheme of isofrequency k-vector curves at both sides of the

boundary between monocrystalline domains of the ultrasonic twinned

structure (left) and for the ‘‘left-handed’’ metal–epoxy composite

(right). The circle corresponds to the isotropic epoxy and the square

corresponds to the anisotropic periodic metallic composite. In the

‘‘left-handed’’ system, the direction of the refracted beam is parallel to

the negative group velocity, vg, of elastic waves above the first band,

just where the numerical experiment is carried out. General size and

shape of isofrequency curves are computed by using the above men-

tioned FDTD program Wave 2000. The scale indicates the size of the

wave vector.

M. Torres, F.R. Montero de Espinosa / Ultrasonics 42 (2004) 787–790 789

times stronger than along the (1 1 0) one. On the other

hand, by comparing wave velocities both in the ultra-

sonic crystal and in the pure aluminum, where is about

6380 m/s, real magnitudes of the velocity and wave

vector can be calculated for each frequency along any

direction.

To visualize such an wave vector anisotropy experi-

mentally [12], we substituted one mercury cylinder by apiezoelectric vibrator impulsively excited. A surface

scanning of the vibration amplitude is shown in Fig. 6,

where we can clearly see that the originally cylindrical

front wave became square after coupling with the lattice,

propagating along the allowed (1 0 0) and (0 1 0) easy

directions. Consequently, Huygens’ envelope of the

secondary waves cannot be properly defined and re-

fraction at a grain boundary of this ultrasonic crystal isof an anomalous extreme nature [12]. On the other

hand, this anisotropic wave propagation can be used to

generate highly directional beams by devices based on

ultrasonic band gaps [12].

In Fig. 7 (left) we schematically show isofrequency k-vector curves in both monocrystalline twinned domains

at both sides of the corresponding boundary. In spite of

the schematic condition of Fig. 7, according to the

above mentioned procedure based on theoretical cal-culation by using the FDTD program Wave 2000, the

general square shape of isofrequency curves and the

magnitude of corresponding wave vector can be com-

puted. Incident and refracted beams are indicated. We

can see that incident waves can be coupled only to the

easy orientation of the neighbor domain. Waves

Fig. 8. Experimental vibration amplitude scanning at the upper surface

of a twinned ultrasonic device. Dotted lines and letters indicate the

boundaries of the twin between the crystallographic domains as shown

in Fig. 4. Scanning amplitudes are also shown (right). The wave

splitting by the ultrasonic wedge is clearly observed. More details can

be found in Ref. [12].

Fig. 9. Theoretically computed pattern showing traces of negative re-

fraction in a potential ‘‘left-handed’’ composite. The device consists of

a slab of ultrasonic crystal (left) attached to an epoxy region (right).

Vertical line indicates the corresponding boundary. The mercury cyl-

inders are indicated by superimposed circles. A plane wave source is

located in the epoxy region forming an acute angle with the boundary.

Arrows indicate how the wave-front advances as the simulation com-

puter experiment is carried out. A apparent behaviour of negative

refraction is shown. The anisotropy of the wave propagation within

the periodic metallic composite could explain the apparent negative

refraction. The wave velocity in the epoxy region is about 2700 m/s. A

great backscattering on the boundary is observed due to the mismatch

in acoustic impedances.

790 M. Torres, F.R. Montero de Espinosa / Ultrasonics 42 (2004) 787–790

propagating along the crystalline domain with (1 0 0),

AA0, orientation are 45� bent at the grain boundary

following an anomalous refraction law which could

mean the threshold of the negative refraction. The

strong wave–lattice coupling causes the strong anomaly

in Snell’s law.

To visualize experimentally the anomalous refraction,

we launched a single frequency (0.875 MHz) plane wavefrom A and the vibration amplitudes of the elastic waves

were scanned at the upper surface of the twinned device

shown in Fig. 4 [12]. In Fig. 8 we show the vibration

amplitude scanning. As an aid to visualize the extreme

refraction at the boundaries, producing 45� bending of

the wave, we use arrows in the mentioned vibration

amplitude image. The vertical arrow indicates the

launching wave direction while lateral arrows indicatethe propagation directions perpendicular to the crystal-

line domain boundaries, i.e., the splitting directions.

A recent theoretical work indicates that negative re-

fraction phenomena in photonic crystals are possible in

regimes of negative group velocity above the first band

near the Brillouin zone center [16]. To show the feasi-

bility of similar phenomena in ultrasonic crystals, we

show here a numerical experiment working also in afrequency range above the first band and the band gap,

namely 0.8–0.9 MHz. By using the FDTD program

Wave 2000, in Fig. 9 we show the anomalous refraction

of a plane wave at the frequency of 0.875 MHz gener-

ated by a wide transducer introduced into epoxy and

launched with an angle to the boundary of a slab of

ultrasonic crystal. Corresponding isofrequency k-vectorcurves are schematically shown in Fig. 7 (right). Therefracted beam is parallel to the negative group velocity,

vg, of elastic waves. Due to the anisotropic character of

the wave propagation within the ultrasonic crystal slab,

the wave goes out of the slab as corresponding to an

example of apparent negative refraction. Ultrasonic

experiments about this ‘‘left-handed’’ composite are in

course [17].

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