JOURNAL OF APPLIED PHYSICS 104, 014909 共2008兲
Experimental determination for resonance-induced transmission
of acoustic waves through subwavelength hole arrays
Bo Hou,1 Jun Mei,1,2 Manzhu Ke,2 Zhengyou Liu,2 Jing Shi,2 and Weijia Wen1,2,a兲
1
Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay,
Kowloon, Hong Kong
2
Department of Physics, Wuhan University, Wuhan 430072, China
共Received 18 December 2007; accepted 27 April 2008; published online 8 July 2008兲
We measured acoustic transmissions through subwavelength hole arrays fabricated on brass plates
at normal and oblique incidence. It is found experimentally that the transmission phenomena for the
hole array in thin plate case are analogous to the previously observed enhanced transmission of
electromagnetic waves 关Ebbesen et al., Nature 共London兲 391, 667 共1998兲兴. While for the hole array
in thick plate case, the transmission peaks of acoustic wave occur well below Wood’s anomalies,
and the spectrum characteristics reveal a Fabry–Pérot-like resonance. An effective fluid approach is
conceived and it can well describe the transmission properties of the hole array in thick plates within
a range of incidence angle. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2951457兴
I. INTRODUCTION
Enhanced transmission of light through a lattice of subwavelength holes fabricated on a metallic film was observed
by Ebbesen and co-workers in 1998,1–8 where the optical
transmission can be much larger than the area fraction of the
holes at specific frequencies. Since then, the remarkable phenomenon has inspired a tremendous amount of attention and
works on resonant transmissions of electromagnetic 共EM兲
waves through various apertures on either metallic or dielectric structure.9–14 Phenomenologically, various observed
transmission resonances are associated with two geometrical
factors: structural factor emerging globally from the lattice
periodicity and aperture factor owned locally by the individual unit.15–18 Structural-factor-related resonances typically
have the transmission wavelength comparable to the lattice
constant and are dependent strongly on the incidence angle.
In sharp contrast, aperture-factor-related resonances have the
wavelength determined mainly by the transversal/
longitudinal dimensions of the aperture and are not sensitive
to the incidence angle.
In this paper, we investigated experimentally the acoustic wave transmissions through a two-dimensional array of
subwavelength holes, considering that acoustic and EM
waves share a lot of wave phenomena. However, they have
something in difference. In nature, acoustic wave is a scalar
longitudinal wave in inviscid fluids, while EM wave is a
vector transverse wave. Consequently, a subwavelength hole
has no cutoff for acoustic wave, but does for EM wave,
which underlies the distinct transmissions of acoustic/EM
waves through a hole in an ideally rigid/conducting screen.
The acoustic transmission approaches a constant, 8 / ␲2,
whereas the EM transmission goes to zero, with decreasing
the ratio d / ␭ 共hole diameter/wavelength兲.19,20
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0021-8979/2008/104共1兲/014909/5/$23.00
Transmission/diffraction by an acoustical grating is an
old problem, and the previous investigations addressed some
cases: one-dimensional periodic slits in a rigid screen,21 a
single hole in a thick wall,22 and a one-dimensional grating
composed of parallel steel rods with finite grating
thickness.23 Here we measured the acoustic transmissions
through a two-dimensional array 共square lattice兲 of subwavelength hole fabricated on either thin or thick brass plates at
normal and oblique incidence. It is found that the acoustic
transmission phenomenon for the hole array in thin plates is
analogous completely to the case of EM wave,1,2 except for
the transmission phase. For the hole array in thick plates, the
transmission peaks are related to the Fabry–Pérot-like 共FPlike兲 resonances inside the holes and can occur to the frequencies well below Wood’s anomalies. An effective medium model is derived, where the material parameters of the
new effective fluid is scaled from those of the hole-filled
fluid by a geometrical value. The effective fluid model may
describe well the transmission properties of the hole array
with large thickness within a range of incidence angle.
II. SAMPLES AND MEASUREMENTS
In our experiments, a square lattice of circular holes with
diameter, d, and lattice constant, a, was drilled on a smooth
brass plate of thickness, t. The schematic picture of the unit
cell is illustrated in the inset of Fig. 1. The measurements of
far field transmissions of acoustic waves in the ultrasonic
frequency regime were performed in a large water tank. Two
immersion transducers were employed as ultrasonic generator and receiver, and the sample was placed at a rotation
stage located between the two transducers at an appropriate
distance. The sample could be rotated so that the oblique
incidences were measured. The ultrasonic pulse was incident
upon the sample and the transmitted signal was collected by
the receiver, collinear with the incident wave. Transmission
magnitude, T, and transmission phase, ␸, of the sample were
104, 014909-1
© 2008 American Institute of Physics
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014909-2
J. Appl. Phys. 104, 014909 共2008兲
Hou et al.
FIG. 1. 共Color online兲 Transmittance of the acoustic waves at normal incidence through a hole array 共solid line兲 and a smooth brass plate 共dashed
line兲 with the same thickness. The hole array has the parameters: the hole
diameter d = 0.5 mm, the lattice constant a = 1.5 mm, and the plate thickness
t = 0.5 mm. A schematic picture of the unit cell is illustrated as the inset. The
level dot line represents the area fraction of holes in the array.
obtained by normalizing the Fourier transformed spectra of
the transmitted signal through the sample, 兩As共f兲兩exp关j␸s共f兲兴,
with respect to the signal through the water background
共without the sample in place兲, 兩Ab共f兲兩exp关j␸b共f兲兴, where f is
the frequency and j2 = −1. Consequently, T = 兩As共f兲 / Ab共f兲兩 and
␸ = ␸s共f兲 − ␸b共f兲 − 2␲ ft / c 共c = 1490 m / s being the speed of
acoustic wave in water兲.24
In this paper, the term “transmission” when referring to
the spectrum means the amplitude ratio, T, of the transmitted
and the incident waves. To figure out whether the transmission is enhanced, we need calculate the ratio of transmitted
energy flow to the energy flow incident on all holes. Within a
unit cell, the ratio can be expressed by
兩At/Ai兩2
T2 T
I ta 2
=
=
= ,
Ii · ␲d2/4 ␲d2/共4a2兲 ␰ ␰
共1兲
where It 共At兲 and Ii 共Ai兲 denote the intensities 共amplitudes兲 of
transmitted and incident acoustic waves, respectively, and ␰
= ␲d2 / 共4a2兲 is the area fraction of the holes. We call the
squared transmission magnitude T2 as transmittance T representing the acoustic intensity transmission. If T / ␰ ⬎ 1, then
the enhanced transmission is obtained. Thus, in the following
discussions we use the transmittance spectrum whenever
comparing with the area fraction of the holes.
III. RESULTS AND ANALYSES
A. The thin plate case
First we measured the acoustic transmission of a hole
array with the parameters d = 0.5, a = 1.5, and t = 0.5 mm. Figure 1 shows the transmittance of the hole array at normal
incidence, compared to the transmittance of a smooth brass
plate with identical thickness. For the smooth brass plate,
very low transmittance is seen because of the acoustic impedance mismatch 共␩brass / ␩water ⬇ 25兲. It is noticed that the
transmittance rises at lower frequencies, which indicates a
thin brass plate can not block acoustic waves of very long
wavelength or very low frequency. This fact is different from
the EM case where a sheet of metal as thin as skin depth
FIG. 2. 共Color online兲 共a兲 Acoustic transmissions at normal incidence of the
hole arrays with the same lattice constant, 2.0 mm, and the same thickness,
0.5 mm, but different diameters as denoted. The measured transmission
phase curve is for the hole array: d = 1.2, a = 2.0, and t = 0.5 mm. 共b兲 Acoustic
transmissions at normal incidence for the hole array: d = 0.5, a = 1.5 and t
= 0.5 mm. The arrows indicate Wood’s anomalies, ␭ = a and ␭ = a / 冑2.
works well. For the hole array, a pronounced peak is seen at
0.85 MHz and followed by a transmittance zero close to
1.0 MHz which is just Wood’s anomaly ␭ = a. The peak has
the transmittance 共68%兲, much lager than the area fraction
共8.7%兲 of holes occupation in the array structure, and shows
an acoustic transmission enhancement through the hole array,
similar to the EM case.1
We also investigated the dependence of the transmission
peak on the lattice constant. Figure 2共a兲 shows the normal
transmissions of the hole arrays with identical lattice constant a = 2.0 mm and different hole diameters. The transmission peak and two Wood’s anomalies 共pointed by arrows兲 are
identified at ⬃0.75 and ⬃1.1 MHz. With the larger diameter
holes 共d = 1.2 mm兲, the peak becomes more pronounced. For
comparison, the transmission curve of the array of a = 1.5
mm is replotted in Fig. 2共b兲. It is clear to show that the peaks
and Wood’s anomalies downshift to lower frequencies as the
lattice constant increases. In Fig. 2共a兲, we also plotted the
measured transmission phase ␸ for the hole array of d = 1.2,
a = 2.0, and t = 0.5 mm, and found ␸ = −0.98␲ at the peak
frequency. The approximate −␲ phase change reveals the
oscillations of the acoustic field on the front and rear surfaces
of the plate are out of phase, which is distinct from the corresponding characteristic in the EM case. For EM wave
transmitted through a hole array, the hole acts as barrier due
to the transmission frequency much lower than the cutoff
frequency of the hole, and the wave has to tunnel through the
hole in a form of evanescent field. So the phase change of the
EM wave across the holey film/plate assumes nearly zero.25
This difference in transmission phase bares the distinct behaviors of a hole to acoustic and EM waves, again.
Figure 3 shows the transmission spectra at oblique incidence measured with the incident angle ␪ varying from 0° to
25° for the hole array of d = 1.2, a = 2.0, and t = 0.5 mm. The
obliquity occurred along the 关1,0兴 direction of the array, as
illustrated in Fig. 3共a兲. The transmission map is plotted as a
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014909-3
J. Appl. Phys. 104, 014909 共2008兲
Hou et al.
FIG. 3. 共Color online兲 dif共a兲 The schematic picture of oblique incidence of
acoustic waves. The wave vector kinc
represents the incident wave illuminating the hole array at the incidence
angle ␪. The obliquity occurs along
the 关1,0兴 direction of the array. The
wave vector k共0,0兲 represents the 共0,0兲order transmitted wave. The wave vector kdiff共l,m兲 represents the 共l , m兲-order
diffraction wave. 共b兲 Acoustic transmission magnitude plotted as a function of the wave frequency and the incidence angle for the hole array, d
= 1.2, a = 2.0, and t = 0.5 mm. The solid
lines superposed are the variation
curves of Wood’s anomalies with the
incidence angle.
function of both the frequency and the incidence angle. The
predicted variation of Wood’s anomalies versus angle is plotted as solid lines and is superposed on the map. Derived
from the conservation of momentum, the variation relation
reads
af 共l,m兲/c = 共l sin ␪ + 冑l2 + m2 cos2 ␪兲/cos2 ␪ ,
共2兲
for the Wood’s anomaly f 共l,m兲 of order 共l , m兲. It is seen from
the map that the measured shifting of Wood’s anomalies with
the incidence angle agrees well with the solid lines. On the
other hand, the peaks exhibit a strong angle-dependent behavior in the same way as Wood’s anomalies.
In recent investigations, it is demonstrated that the structure factor 共SF兲 resonance can be responsible for enhanced
transmissions of EM waves through subwavelength hole
arrays.15,16 However, for the role of any surface wave, supported by the grating medium, playing in the enhanced optical transmission process, there are different views.26–28. We
have concluded that the acoustic surface wave at the brasswater interface might play no role in the present transmission
phenomenon, and shown that the SF resonance holds for
acoustic waves by generalizing the proof of EM waves.29,30
The SF resonance has some spectral features: the resonant
wavelength is determined essentially by the lattice constant
and is very sensitive to the incidence angle with accompanied by Wood’s anomalies. Here, the experimental results for
the enhanced acoustic transmission through the hole array in
the 0.5 mm thick plate manifests the features of SF resonance.
B. The thick plate case
When the plate thickness becomes larger, the situations
begin to divide for two types of waves. For EM wave, the
transmission peak will diminish after the metallic film/plate
becomes thick enough because the holes have the cutoff. In
sharp contrast, there is no cutoff for acoustic waves to propagate through the holes. When the thickness is large enough,
for instance t = 2.3 mm, there can be multiple transmission
peaks well below the Wood’s anomaly, as shown in Figs.
4共b兲 and 4共c兲. Their spectra show the typical characteristics
of FP resonance. In fact, these transmission peaks are caused
by standing-wave-formed resonances of the acoustic wave
establishing inside the hole channel. However, these resonances undergo a tuning, to some degree, by diffractive evanescent waves parasitical to a grating, and consequently deviate from the ordinary FP conditions while the plate
thickness becomes comparable to the lattice constant.29 Under the limit of thick plate 共the ratio a / t → 0兲, they coincide
with the FP resonance conditions.
For a very small a / t ratio, these resonance wavelengths
are much larger than the lattice constant, allowing us to take
a view of effective media. Here we employ a simple argument in the same fashion as the EM case31 with the assumption of brass plate being rigid, and find that the structure of
an array of holes fabricated in a rigid plate and filled with a
fluid 共mass density ␳0 and bulk modulus ␬兲 may be viewed
as an effective fluid with the same thickness, effective mass
density ¯␳0 and bulk modulus ¯␬, as shown schematically in
Fig. 4共a兲. It is known that the acoustic wave is characterized
by the pressure field, p, and velocity field, u. Averaging the
pressure field in the holes, we get the effective pressure field
in the effective fluid p̄ = ␰ p. Requiring the acoustic energy
flow across the surface to be the same for the hole array and
the effective fluid, i.e., pu · ␲d2 / 4 = p̄ū · a2, we obtain the effective velocity ū = u. Also the total acoustic energy for both
systems are required to be the same, 共 21 ␳0u2 + 21 p2 / ␬兲␲d2 / 4t
= 共 21 ¯␳0ū2 + 21 p̄2 / ¯␬兲a2t, which gives us the effective parameters
¯␳0 = ␰␳0 and ¯␬ = ␰␬. Thereafter, the acoustic speed and impedance of the effective fluid are ¯␯ = ␯ = 冑␬ / ␳0 and ¯␩ = ␰␩
= ␰冑␳0␬, where ␯ and ␩ are the acoustic speed and impedance of the filling fluid, respectively. The relations indicate
the acoustic speed remains unchanged and the impedance is
scaled by a factor of the area fraction of holes for the effective fluid. It is noted that an analytical derivation also produces the similar conclusion.32
The above argument is applicable under long wavelength
limit 共a / ␭ → 0兲 where diffractive evanescent waves are negligible. For the sample with thickness comparable to lattice
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014909-4
J. Appl. Phys. 104, 014909 共2008兲
Hou et al.
FIG. 4. 共Color online兲 共a兲 A hole array
drilled in a rigid body of thickness t
and filled by a fluid with mass density
␳0 and bulk modulus ␬ can be modeled as an effective fluid with the same
thickness t, the mass density ¯␳0, and
the bulk modulus ¯␬. 共b兲 Normal transmission of acoustic waves through the
hole array 共open circles兲, d = 1.2, a
= 2.0, and t = 2.3 mm, immersed in water and through the effective fluid
layer 共solid line兲 immersed in water
alike. Open circles are the measurement data and the solid lines are the
calculation employing the effective
parameters n̄ = 1.19 and ¯␩ = 0.28␩water.
共c兲 Normal transmission of acoustic
waves through the hole array 共open
circles兲, d = 0.5, a = 2.0, and t
= 2.3 mm, immersed in water and
through the effective fluid layer 共solid
line兲 immersed in water alike. Open
circles are the measurement data and
the solid lines are the calculation employing the effective parameters n̄
= 1.19 and ¯␩ = 0.05␩water.
constant, the diffractive evanescent waves tune the FP resonances and the resultant transmission resonances can occur
for channel length ⬃16% thinner than required by the FP
resonances.29 Superficially, this diffractive effect is substitutable by a slowing of acoustic wave propagation inside the
holes, i.e., ␯ = 0.84c. Based on the above argument, the effective fluid equivalent to a 2.3 mm thick sample has the acoustic speed ¯␯ = ␯ = 0.84c, or the acoustic refractive index n̄
= c / ¯␯ = 1.19, and the effective impedance ¯␩ = ␰␩ = ␰␩water.
With these two effective parameters at hand, we calculated
the transmission spectra, both magnitude and phase, of the
effective fluid layer at normal incidence according to the
formula
T exp共j ␾兲 =
冉
1
冊
¯␩
␩water
j
cos共k0n̄t兲 +
+
sin共k0n̄t兲
2 ␩water
¯␩
,
共3兲
where k0 = 2␲ f / c is the wavenumber of the incidence wave.
The calculated results 共solid lines兲 are shown in Figs. 4共b兲
and 4共c兲 to compare to the experimental data 共open circles兲
for two samples with identical thickness of 2.3 mm and identical lattice constant of 2.0 mm but different hole diameters
of 1.2 and 0.5 mm. Good agreement between the calculations and the experiments is seen at the frequencies below
Wood’s anomaly 共0.75 MHz兲, which verifies the applicability of the effective fluid model at normal incidence.
Likewise, we measured the transmission of the hole array, d = 1.2, a = 2.0, and t = 2.3 mm, at oblique incidence. The
obliquity occurred along the 关1,0兴 direction of the array. Figure 5 shows the measured results and the angle-dependent
FIG. 5. 共Color online兲 Acoustic transmission magnitude plotted as a function of the wave frequency and the incidence angle for the hole array d
= 1.2, a = 2.0, and t = 2.3 mm. The obliquity occurs along the 关1,0兴 direction
of the array. The solid line superposed are the variation curve of Wood’s
anomaly 共−1 , 0兲 with the incidence angle. The open circles superposed denote the variation of the transmission peaks, calculated from the FP resonance condition of the effective fluid layer at oblique incidence.
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014909-5
variation of Wood’s anomaly 共−1 , 0兲 共solid line兲. Two flatbands appear below 0.75 MHz and they are the two peaks,
the first and second FP resonances, in Fig. 4共b兲 where the
phase values indicate the order of FP resonances. The open
circles superimposed are the variations of the transmission
peaks of the effective fluid layer which are obtained from the
FP
resonance
condition
at
oblique
incidence,
sin关k0n̄t冑1 − 共sin ␪ / n̄兲2兴 = 0.
The agreement between the calculated variations of the
transmission peaks and the measured results indicates the
applicability of the effective fluid model persists to a range
of incidence angle. The discrepancy at ␪ ⬎ 15° for the second
FP transmission peak is due to the emerging of the 共−1 , 0兲
diffraction order nearby, and the red flatband is seen to terminate upon crossing with Wood’s anomaly 共−1 , 0兲 in the
figure. Where the cross happens, there will be the strong
coupling interaction between SF resonance of the array and
FP-like resonance localized at each hole, which possibly
gives rise to the 0.48 MHz band at ␪ = 25°.
IV. CONCLUSION
We investigated experimentally the acoustic transmission through the subwavelength hole arrays fabricated on
brass plates at normal and oblique incidence within ultrasonic frequencies regime. The transmission phenomena for
the hole array in thin brass plates, analogous to the observed
enhanced transmission of EM waves through subwavelength
hole arrays in a metallic film,1,2 exhibit the transmission enhancement 共transmittance being larger than the area fraction
of the holes兲. At the peak frequency, the transmission phase
is nearly −␲, indicating the out-of-phase oscillations of the
acoustic field at two surfaces of the plate. However, for the
hole array in thick brass plates, the transmission peaks of
acoustic waves are related to the FP-like resonances inside
the holes and therefore occur well below Wood’s anomaly
since a hole has no cutoff frequency for acoustic propagation. Consequently, the structure can be viewed as a new
fluid with effective mass density and bulk modulus scaled,
under long wavelength limit, by a factor of area fraction of
the holes. The effective medium model describes well the
transmission properties of the hole array within a range of
incidence angle.
ACKNOWLEDGMENTS
This work was supported by NSFC/RGC joint research
Grants No. NគHKUST632/07 and 10731160613.
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J. Appl. Phys. 104, 014909 共2008兲
Hou et al.
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