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PRL 102, 144301 (2009)
PHYSICAL REVIEW LETTERS
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Angle-Dependent Ultrasonic Transmission through Plates with Subwavelength Hole Arrays
Héctor Estrada,1,2 F. Javier Garcı́a de Abajo,3,* Pilar Candelas,1 Antonio Uris,1
Francisco Belmar,1 and Francisco Meseguer1,2,†
1
Centro de Tecnologı́as Fı́sicas, Unidad Asociada ICMM-CSIC/UPV, Universidad Politécnica de Valencia,
Avenida de los Naranjos s/n. 46022 Valencia, Spain
2
Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain
3
Instituto de Óptica—CSIC and Unidad Asociada CSIC-Universidade de Vigo, Serrano 121, 28006 Madrid, Spain
(Received 7 May 2008; revised manuscript received 10 February 2009; published 8 April 2009)
We study the angle and frequency dependence of sound transmission through water-immersed
perforated aluminum plates. Three types of resonances are found to govern the acoustic properties of
the plates: lattice resonances in periodic arrays, Fabry-Perot modes of the hole cavities, and elastic Lamb
modes. The last two of them are still present in random arrays and have no parallel in optical transmission
through holes. These modes are identified by comparing experiment with various levels of theoretical
analysis, including full solution of the elasto-acoustic wave equations. We observe strong mixture of
different transmission mechanisms, giving rise to unique acoustic behavior and opening new perspectives
for exotic wave phenomena.
DOI: 10.1103/PhysRevLett.102.144301
PACS numbers: 43.35.+d, 42.79.Dj
The similarities and discrepancies between mechanical
waves, such as sound, and electromagnetic waves have
puzzled scientists for a long time [1]. This is much more
than a semantic question. It deals with the essential nature
of matter, because sound is basically a classical-physics
phenomenon, while light is deeply rooted into the
quantum-mechanical description of reality. Light and
sound have been confronted in numerous experiments
and theories, ranging from band gap effects in photonic
[2] and phononic [3] crystals to negative refraction [4] and
invisibility [5] phenomena.
The discovery of extraordinary optical transmission in
metallic membranes perforated by subwavelength apertures [6] has raised again the question of the similarity
between sound and light in the context of wave transmission. Very recently, several groups have reported on the
transmission properties of sound through plates with slits
[7,8] and holes [9–11]. Some groups [7–9] claim that
extraordinary transmission of sound is similar to its optical
counterpart in corrugated metal films. However, Hou et al.
[10] have shown that, despite some similarities, intrinsic
differences separate light and sound. Holes in membranes
cannot sustain optical modes in the subwavelength regime.
This is not the case for sound. Moreover, we have recently
shown that sound has specific properties unforeseen from
the perspective of optical transmission in metallic films
[11]. First, extraordinary acoustic transmission is not
solely determined by hole periodicity, since it is also
observed in plates pierced by a random distribution of
holes. More importantly, periodically perforated plates
are capable of shielding sound in a wide region near the
onset of diffraction much better than what is predicted by
the well-known mass law [11]. This behavior, with no
parallel in optics, should have important applications in
soundproofing engineering. However, a detailed study is
0031-9007=09=102(14)=144301(4)
still missing regarding the role of the wave vector parallel
to the plate in sound transmission for oblique incidence in
both periodic and random distributions of holes.
In this Letter, we investigate the angular dependence of
sound transmission through water-immersed aluminum
plates perforated by subwavelength apertures. Our results
reinforce previous findings for normal-incidence transmission [11] but otherwise show a remarkable interplay between Lamb waves and lattice modes induced by periodic
distributions of holes, which is unveiled only under oblique
incidence. A rich structure is observed in the measured
angle and frequency distribution of transmission intensity,
which is correctly described by numerical solution of the
full elasto-acoustic wave equations. Partial understanding of the observed features comes from comparison of
the full theory with (i) a description of the plate as a hard
solid, leading to lattice resonances and the resulting extraordinary transmission features, and (ii) a real-solid
homogeneous-plate description yielding the Lamb modes
of the plate. However, both experiment and full real-solid
theory for perforated plates reveal complex hybridization
of different resonance mechanisms, leading to numerous
mode anticrossings, enhancement of some of the lattice
resonances, and a significant shift of the Lamb modes.
Our experimental setup is based on the well-known
ultrasonic immersion transmission technique [11]. The
angle of incidence is varied by rotating the sample plate
from 0 to 60 from normal incidence in steps of 1 . The
plates used in this work are made of aluminum (density
¼ 2:7 g=cm3 , longitudinal wave velocity cl ¼
6500 m=s, and transversal wave velocity ct ¼ 3130 m=s
[12]), mechanically drilled (holes of diameter d ¼ 3 mm,
periodically distributed in a squared array of period a ¼
5 mm), and immersed in water. The measured transmission
spectra are collected as a function of frequency ! and wave
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Ó 2009 The American Physical Society
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PHYSICAL REVIEW LETTERS
PRL 102, 144301 (2009)
vector kk parallel to the X direction of the plate [see
Fig. 1(a)]. The frequency, angle, and parallel wave vector
are related through kk ¼ k sin, where k ¼ !=c and c ¼
1480 m=s is the speed of sound in water.
Figures 1(b)–1(d) show transmission intensity maps as a
function of normalized parallel wave vector kk a= and
reduced frequency !a=2c for perforated plates of thickness t ¼ 1, 2, and 5 mm, respectively. A complex structure
of transmission features can be observed. Their evolution
with t allows us to classify them into two distinct categories: (i) The position of some of the features are relatively
insensitive to t, and these are connected to lattice resonances, as we show below; (ii) other features exhibit
dramatic variations with t, and these are related to intrinsic
elastic modes of the film. The presence of mode anticrossings in the intersections between the two types of features
indicates strong mutual coupling.
We have solved the full elasto-acoustic wave equations
[13] under the same conditions as in the actual sampled
plates, for an infinite number of holes. The calculated
results are presented in Figs. 1(e)–1(g). The method of
solution involves the following steps: (i) The displacement
field u and the Lamé coefficients are Fourier transformed
along directions parallel to the periodic plate; (ii) the eigenstates of a 2D crystal formed by infinitely long holes
with the same periodicity as the
P plate are obtained by
inserting the solutions uq ðrÞ ¼ G uq;G exp½iðq þ GÞ r
into the wave equations, where q ¼ ðkk ; qz Þ, the sum runs
over 2D reciprocal-lattice vectors G, and the wave vector
along the holes qz is an eigenvalue of the resulting
quadratic-algebra equations for fixed frequency [14];
(iii) Rayleigh expansions are used for the pressure in the
water outside the plate, whereas the displacement field
inside is expanded in terms of uq eigenstates; (iv) the
continuity of the displacement and the stress in the plate
boundaries leads to a set of linear equations that are solved
to yield the coefficients of these expansions. This method
of solution gives a rigorous expansion for finite plates, in
which the thickness enters through the boundary conditions matching the internal 2D modes to the Rayleigh
expansions outside the film. The calculated transmission
maps [Figs. 1(e)–1(g)] are in reasonable agreement with
experiment. However, it should be mentioned that some
spurious modes appear when a finite number of G’s is used
(1000 in Fig. 1). These modes are mainly localized in the
water-aluminum interface and originate in unphysical values of the Fourier-expanded Lamé coefficients because of
the large mismatch in both media. Spurious modes produce noise that can be partially removed by eliminating
them from the expansion of the displacement u, although
part of their effects are still discernible in Fig. 1 and
increase with t.
Further insight into the physical origin of the transmission features is provided by two simpler models, as shown
in Fig. 2 for t ¼ 2 mm. (i) First, we have performed
calculations in the hard-solid limit [Fig. 2(b)], which is
equivalent to the perfect conductor idealization in electromagnetism. In this limit, the wave equations become scalar
in terms of the pressure field , which satisfies Helmholz’s
equation ðr2 þ k2 Þ ¼ 0 in the water and is subject to the
condition that its normal derivative vanishes at the interface with the solid. This approximation leads to excellent
results when the sound velocity mismatch between solid
and water is large (for example, in brass [11]). This model
produces transmission maxima at slightly lower frequencies with respect to the onset of diffraction (i.e., when a
Bragg beam G becomes grazing [15]), which under the
conditions of Fig. 1(a) leads to
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jkk þ Gj ¼ ðkk þ 2n=aÞ2 þ ð2m=aÞ2 ¼ !=c: (1)
(b)
(c)
(d)
1.1
(a)
π
ω
1
0.9
0.8
0.7
=1 mm
exp.
0.6
1.2
=2 mm
exp.
(e)
=5 mm
exp.
(f)
(g)
1.1
2
Sound Transmission | |
π
ω
1
0.9
0.8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
= 3 mm
= 5 mm
0.7
0
0.5
1
π
1.5
2
=5 mm
theo.
=2 mm
theo.
=1 mm
theo.
0.6
0
0.5
1
π
1.5
2
0
0.5
1
π
1.5
2
FIG. 1 (color online). Angle dependence of sound transmission through aluminum plates immersed in water and perforated by holes
of fixed diameter d ¼ 3 mm, arranged in a square array of period a ¼ 5 mm. (a) Schematic representation of a drilled plate, the
incident sound wave vector k, and its component parallel to the film kk . (b)–(d) Measured sound transmission as a function of kk and
frequency ! for three different film thicknesses t, as indicated by text insets. (e)–(g) Theoretical counterpart of (b)–(d), obtained by
solving the full elasto-acoustic wave equations. The transmission is represented in linear gray scale.
144301-2
ωa/2πc
1
(-2,0)
(0,1)
(-1,1)
0.9
0.8
(-1,0)
0.7
0.6
(c)
(d)
1.1
ωa/2πc
A0
0.9
S0
0.8
0.7
0.6
0
0.5
1
1.5
2
0
0.5
1
k a/π
Sound Transmission |T|
−25
1.5
2
k||a/π
||
−20
−15
−10
2
−5
FIG. 2. Measured transmission and full-theory calculation
compared with hard-solid and homogeneous-plate models for
thickness t ¼ 2 mm. (a) Dependence of the measured transmission intensity on parallel wave vector kk and frequency !
for a water-immersed perforated aluminum film of period a ¼
5 mm and hole diameter d ¼ 3 mm. (b) Calculated transmission
in a hard-solid model [11]. (c) Calculated transmission for a
homogeneous aluminum plate of the same thickness [12].
(d) Numerical solution of the full elasto-acoustic wave equations
for the actual perforated aluminum plate. The transmission is
represented in dB gray scale.
The transmission vanishes right when this condition is
satisfied. Different values of the Miller indices ðn; mÞ
produce the dashed curves of Figs. 2(a), 2(b), and 2(d).
Like in the optical case, these transmission resonances are
driven by lattice-sum singularities originating in cumulative in-phase scattering among the holes of the array
[15,16]. For sufficiently thick samples, Fabry-Perot resonances are also predicted to play a dominant role [11].
(ii) Another insightful approach consists in examining
homogeneous films [Fig. 2(c)], the transmission properties
of which are dominated by the presence of Lamb modes
[12]. These resonances reduce to S0 and A0 for t ¼ 2 mm,
and, in particular, the A0 mode is mixed with a ScholteStoneley mode near grazing incidence. This is reasonable
because this mode (unlike Lamb oscillations) is confined to
the surface in a similar way as surface-plasmon polaritons
in metals.
The measurement in the t ¼ 2 mm drilled plate
[Fig. 2(a)] resembles the hard-solid theory [Fig. 2(b)].
Both of them show transmission dips at the onset of
diffraction [Eq. (1)] and transmission maxima at slightly
lower frequency, similar to what happens in optical transmission as a result of the interplay between finite hole
polarization and divergent interhole interaction [16].
However, a new mode consistent with S0 appears and
strongly interacts with the lattice resonances at kk a= 0:6 and 1.0 in the experiment. This interaction is well
reproduced by the full wave calculation [Fig. 2(d)]. The
Lamb mode crosses the ð1; 0Þ lattice-sum singularity,
meets the crossing of the ð0; 1Þ and ð1; 1Þ features at
the boundary of the first Brillouin zone, where it has zero
group velocity, as predicted by the two-band model [17],
and then falls down with negative group velocity outside
that zone. A similar behavior is observed for thinner plates
[Figs. 1(b) and 1(e)]. As a rule, the low-frequency S0 mode
of the perforated plates is significantly less steep than in
nonperforated plates [cf. Figs. 2(a) and 2(c)], implying
smaller group velocity (i.e., the effective holey-plate parameters correspond to smaller Lamé coefficients compared to the homogeneous aluminum plate and therefore
lower elastic wave velocities).
The interaction between lattice resonances and Lamb
modes becomes more involved when the thickness increases, as shown in Figs. 1(d) and 1(g) for t ¼ 5 mm. In
homogeneous aluminum plates of the same thickness (see
auxiliary material [18]), the A0 Lamb mode moves to lower
wave vector values, away from the Scholte-Stoneley mode,
while the A1 Lamb mode appears in our range of measurement. As a result, the S0 mode is much fainter in the t ¼
5 mm perforated plate compared to plates of smaller thickness, but we observe instead a rich elastic-mode structure
with high transmission values above the ð1; 0Þ latticesum singularity. Furthermore, the near-normal-incidence
measured transmission in the pierced thick plate [Fig. 1(d)]
exhibits a peak originating in a Fabry-Perot resonance,
flanked by a lower lattice resonance, as inferred by comparison with hard-solid theory [11]. Also, the transmission
features become broader in thicker plates.
(a )
(b)
1.1
1
π
(b) (1,0)
0.9
ω
(a)
1.1
1
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PHYSICAL REVIEW LETTERS
PRL 102, 144301 (2009)
0.8
0.7
0.6
0
0.5
1
π
1.5
2
0
0.5
1
π
1.5
2
Sound Transmission
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
(c)
(d)
2
1
0.03
0
2
1
0.03
0
−1
0.01
−2
−1
0.01
−2
−2
0
2
−2
0
2
FIG. 3. Transmission through disordered hole arrays compared
to ordered arrays. (a) Dependence of the measured transmission
intensity on parallel wave vector kk and frequency ! for a waterimmersed perforated aluminum film of thickness t ¼ 5 mm,
period a ¼ 5 mm, and hole diameter d ¼ 3 mm. (b) Measured
transmission for an array of randomly distributed holes of the
same thickness and diameter and with the same filling fraction of
the holes in the plate. (c),(d) 2D Fourier transforms (contour
plots in log scale) of the film openings for the (c) periodic and
(d) random arrays.
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PRL 102, 144301 (2009)
PHYSICAL REVIEW LETTERS
The dramatic influence of hole ordering on the transmission performance of perforated plates is demonstrated
in Fig. 3, which shows measurements for 2 mm thick plates
pierced by periodic [Fig. 3(a), period a ¼ 5 mm] and
random [Fig. 3(b)] arrangements of holes (diameter d ¼
3 mm) with the same average filling fraction. The rich
interplay between lattice modes and intrinsic plate modes
in the ordered array [Fig. 3(a)] is completely absent in the
random sample. However, the latter displays a feature
resembling the S0 Lamb mode of the homogeneous plate
[cf. Figs. 2(c) and 3(b)]. Interestingly, the Fourier transform of the 2D distribution of geometrical openings in
random arrays shows a broad annular maximum with a
radius close to a=, which gives rise to a broad dark region
near normal incidence, close to the lattice resonance of the
ordered array [19]. The phase velocity of the lowfrequency S0 mode is found to be 3150 m=s in both
perforated plates, which seems to respond to the average
elastic parameters of the water-aluminum holey sample as
noted above, quite different from the faster S0 mode of the
pure aluminum plate [5486 m=s, Fig. 2(c)]. In contrast to
random arrays, the phononic crystal generated for the
Lamb waves in periodic arrangements [20] could interact
with Fabry-Perot resonances and lattice-driven modes,
although further research is needed to address this issue.
In summary, the angle-resolved acoustic transmission
through perforated plates exhibits both lattice resonances,
which are similar in nature to their extraordinary optical
transmission counterpart, and excitation of intrinsic elastic
Lamb modes of the plates. The latter constitute a genuine
aspect of sound transmission. These two types of modes
interact with each other and produce the complex transmission patterns reported here through both measurements
and full solution of the elasto-acoustic wave equations
(Fig. 1). The nature of the modes becomes clear when
comparing these results with the calculated transmission
of either a homogeneous plate, dominated by Lamb modes,
or a hard-solid drilled plate, showing lattice resonances
(Fig. 2). The interplay between lattice and intrinsic resonances, their interaction with Fabry-Perot modes of the
hole cavities, and the partial transparency of real materials
to sound define altogether a new scenario in wave transmission through subwavelength apertures and open up a
source of novel phenomena and applications of the field of
sound transmission through real materials structured at
subwavelength scales.
This work has been supported by the Spanish MCeI
(MAT2006-03097, MAT2007-66050, and NanoLight.es)
and the EU (NMP4-SL-2008-213669-ENSEMBLE).
H. E. acknowledges support from CSIC-JAE.
†
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
*Corresponding author.
[email protected]
144301-4
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10 APRIL 2009
Corresponding author.
[email protected]
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Abajo, and F. Meseguer, Phys. Rev. Lett. 101, 084302
(2008).
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69, 064301 (2004).
The wave equation contains both linear and quadratic
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problem.
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See EPAPS Document No. E-PRLTAO-102-020916 for
auxiliary material. For more information on EPAPS, see
http://www.aip.org/pubservs/epaps.html.
Like in optics, the strength of the hole polarization is
proportional to p ¼ 1=ð1 GÞ, where is the hole
polarizability and G is the sum of the interaction of each
hole with all other holes in the array [16]. The real part
of the sum G is usually divergent under the conditions of
Eq. (1) in ordered arrays, and this is directly related to the
divergence of the Fourier transform of the hole distribution [Fig. 3(c)], whereas the imaginary part is partially
compensated by 1 . In random arrays, G has broader,
smaller maxima, accompanied by a significant imaginary
part, and still leading to small values of p [i.e., depleted
transmission in Fig. 3(b)] but not to enhanced transmission.
X. Zhang, T. Jackson, E. Lafond, P. Deymier, and
J. Vasseur, Appl. Phys. Lett. 88, 041911 (2006).
Angle-dependent ultrasonic transmission through plates with
subwavelength hole arrays
- AUXILIARY MATERIAL Héctor Estrada,1, 2 F. Javier Garcı́a de Abajo,3, ∗ Pilar Candelas,1
Antonio Uris,1 Francisco Belmar,1 and Francisco Meseguer1, 2, †
1
Centro de Tecnologı́as Fı́sicas, Unidad Asociada ICMM- CSIC/UPV,
Universidad Politécnica de Valencia,
Av. de los Naranjos s/n. 46022 Valencia, Spain
2
Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain
3
Instituto de Óptica - CSIC and Unidad Asociada
CSIC-Universidade de Vigo, Serrano 121, 28006 Madrid, Spain
(Dated: February 10, 2009)
1
The interaction between lattice resonances of the hole array and the intrinsic modes of
the plate is very complex. For thin plates, the lattice resonances predicted by the hardsolid model can be clearly identified in the observed spectra. However, thicker plates show
strong hybridization with intrinsic acoustic modes of the plate (Lamb and Scholte-Stoneley
modes), which can give rise to strong transmission. In particular, Scholte-Stoneley modes are
fluid-solid interface modes similar to surface plasmon polaritons for electromagnetic waves.
We show these modes in Fig. 1 for homogeneous aluminum plates as a function of parallel
wavevector kk and frequency ω, normalized to the thickness t and calculated from a simple
fluid-solid-fluid model [1].
Within the range of measurement in our experiments, only S0 , A0 , and A1 modes are
observed. For thin plates (t = 2 mm and t = 3 mm), only S0 and A0 show up (see Figs.
1 and 2 of the main paper). However, the A1 mode is also present in the thicker plate
(t = 5 mm, see Fig. 3 of the main paper). For this low frequency limit, the phase velocity
of the S0 mode is given by [2]
s
cp = 2ct
1−
c2t
c2l
where ct and cl are the bulk transversal and longitudinal acoustic phase velocities, respec-
4
0.9
3.5
A2
0.8
3
0.7
ωt/2πc
2.5 S2
2
1.5
0.6
S1
0.5
0.4
A1
0.3
1
S0
A0
0.5
0
0.2
Scholte-Stoneley
0.5
1
0.1
1.5
2
k t/2π
||
FIG. 1: Transmission calculations (gray linear scale) for a homogeneous aluminum plate as a function of parallel wavevector kk and frequency ω, both of them normalized with the plate thickness
t. The white dotted line represents the Scholte-Stoneley mode.
2
b
1.1
1.1
1
1
ωp/2πc
ωp/2πc
a
0.9
0.8
0.7
0.9
0.8
0.7
0.6
k||
0
k
θ
0.5
0.6
1
k p/2π
-12 -10 -8
-6 -4
0
0.5
1
k p/2π
||
||
-25 -20 -15 -10 -5
-2
FIG. 2: Transmission (gray scale in dB) as a function of parallel wavevector kk and frequency ω
for a homogeneous aluminum plate of thickness t = 2 mm. (a) Experimental results. (b) Theory.
We take p = 5 mm to facilitate comparison with the perforated plate in the main paper. Notice
that the angular range of measurement does not reach grazing angles, so that the black region in
the lower-right area of the plots (evanescent region) is larger in the experiment than in the theory.
tively. For an aluminum plate with cl = 6500 m/s and ct = 3130 m/s, the velocity of the
S0 mode (also called plate velocity mode) turns out to be cp = 5486 m/s. Figure 2 in this
supplementary information shows the very good agreement between the calculated and the
measured transmission observed for the homogeneous plate. The shift of the features to
lower frequencies can be attributed to the finite size of the plate.
∗
Corresponding author: [email protected]
†
Corresponding author: [email protected]
[1] T. Kundu, Ultrasonic Nondestructive Evaluation (CRC Press LLC, 2004).
[2] D. Royer and E. Dieulesaint, Elastic Waves in Solids, I (Springer-Verlag, 2000).
3