Adv. Studies Theor. Phys., Vol. 6, 2012, no. 1, 19 - 25
Longitudinal and Transverse Elastic Waves in
One-Dimensional Phononic Crystals
B. Manzanares-Martinez, L. Castro-Arce, J. Avila-Diaz
Division de Ciencias e Ingenieria, Unidad Regional Sur, Universidad de Sonora
Boulevard Lazaro Cardenas 100, Navojoa, Sonora 85880, Mexico
P. Castro-Garay, E. Urrutia-Banuelos and J. Manzanares-Martinez1
Departamento de Investigacion en Fisica de la Universidad de Sonora
Apartado Postal 5-088, Hermosillo, Sonora 83190, Mexico
Abstract
We have fabricated one-dimensional phononic crystals making a manual arrangement of alternate layers of aluminum and epoxy. The experimental realization is designed to present longitudinal and transverse
phononic band gaps in the range of megahertz frequencies. Detailed
studies of mechanical waves transmission have been experimentally performed using the Fourier transform of short duration pulses. The band
structure of the phononic crystal is calculated using an analytical formula. There is a good agreement between the experimental results and
the theoretical dispersion relation.
Keywords: Phononic crystal, elastic wave, longitudinal, transverse.
1
Introduction
Phononic crystals are periodic structures made of two materials with different
elastic properties [1]. Phononic crystals with band gaps for mechanical waves
are counterparts -by analogy- to photonic crystals with forbidden gaps for
photon propagation [2]. Phononic crystals are one-, two- or three-dimensional
arrangements with periodicity on the same order of magnitude with the gap
wavelength. In these systems, the main consequence of the periodicity is the
creation of phononic band gaps that are frequency intervals over which the
propagation of mechanical waves is forbidden. Phononic crystals can be used
as isolators, perfect acoustic mirrors, noise suppression or by the introduction
1
[email protected]
20
B. Manzanares-Martinez et al
Figure 1: Schematic of a one-dimensional phononic crystal. (a) The infinite
crystal composed by the repetition of a unit cell of width d. (b) The finite
crystal of width D = 4d + dal in a through transmission experimental setup.
of defects, they can be used as resonators, cavities or guidance of elastic waves.
Phononic crystals in the megahertz (MHz) range are suitable for applications
in biomedical ultrasound or acoustic microscopy [3].
Recently special attention has been paid to the study of one-dimensional
phononic crystals [4-12]. One of the authors has reported the experimental
verification of an omni-directional mirror in one-dimensional phononic crystals
in the range of kHz [8]. More recently have been reported the fabrication of
one-dimensional phononic crystals with periods in the range of the micrometer
where is possible to attain gaps in the gigahertz range [12]. Independently of
the frequency range of the band gaps, to study the mechanical propagation
in these periodic structures is of great interest since new effects related to
the spatial periodicity have been recently reported, such as the modeling of
heterostructures with giant photonic band gaps [13,14], the existence of surface
Tamm states [15,16] or the observation of phononic Bloch oscillations [17,18].
In this work, we present the experimental fabrication and characterization
of a one-dimensional phononic crystal. The transmission spectra shows evidence of the first twelve longitudinal bands, which are in the low megahertz
range. We have also measured some transverse bands. We have performed
theoretical calculations of the band structure using an analytical formula. We
have found an excellent agreement between the experimental results and the
theoretical dispersion relation.
21
Longitudinal and transverse elastic waves
2
Theory
The band structure is calculated for an infinite one-dimensional phononic crystal illustrated in Fig. 1(a). This structure is composed by the repetition of a
unit cell of period d composed by two thin films of aluminum (dark gray) and
epoxy (light gray) of width dal and dep , respectively. The phononic dispersion
relation can be obtained from the analytical formula [12]
!
!
"
#
!
ρep ct,l
2πγdal
2πγdep
1 ρal ct,l
2πγdal
2πγdep
ep
al
cos(kd) = cos
cos
−
+
sin
t,l
t,l
t,l
t,l sin
t,l
2 ρep cep ρal cal
cal
cep
cal
ct,l
ep
(1)
where k is the phononic wave vector, γ is the phonon frequency, ct,l
al and
t,l
cep are the constituent layer longitudinal or transverse phonon velocities, and
finally ρal and ρep are the densities for the aluminum and epoxy, respectively.
3
Longitudinal waves
Fig. 1(b) shows a schematic draw of a finite multilayer of aluminum (5052H32) and epoxy (Epotek 302). The multilayer is composed of five layers of
aluminum and four layers of epoxy. The aluminum layer has a width of dal = 3
mm and the epoxy layer has a width of dep = 4.5 mm. The multilayer period
is d = dal + dep = 7.5 mm. The width of the multilayer is D = 4d + dal . The
incidence and transmission media are illustrated in black at both ends of the
multilayer and are made of high temperature bakelite delay line. The sample
was fabricated by a careful manual arrangement of the subsequent layers of
aluminum and epoxy. We have take care to avoid to produce air bubbles in
the epoxy during the fabrication process. However, is inevitable to produce
a certain amount of air bubbles. Anecdotally, we have found that all our
attempts to remove the air bubbles have produced more air bubbles.
The constituent layer densities for the aluminum and epoxy are ρal = 2680
kg/m3 and ρep = 1100 kg/m3 , respectively. The longitudinal velocities are
clal = 6234 m/s and clep = 2500 m/s. The comparison for the theoretical
dispersion relation (left side) and the experimental transmission (right side)
are presented in Fig. 2. In the right side of panels (a), (b), (c) and (d) we
present the results obtained for the transmission (in arbitrary units, au) for
the transducers of 250 kHz, 500 kHz, 1 MHz and 1.5 MHz. We observe that in
all the cases, it exist a good agreement between the high transmission ranges
with the allowed bands. Conversely, exist a clear relation between the low
transmission zones and the band gaps. It is important to note that in panel
(d) we verify the existence of the 11th and 12th bands which is a prove of the
good crystalline quality of the sample.
!
22
B. Manzanares-Martinez et al
k (2
0.0
z
/d)
k (2 /d)
0.5
z
0.0
0.5
300k
150k
0.00
k (2
0.0
(b)
Frequency (Hz)
Frequency (Hz)
(a)
z
/d)
350k
0.01
0.00
Transmission (au)
0.5
0.0
800.0k
0.5
(d)
2.0M
Frequency (Hz)
1.2M
z
0.04
Transmission (au)
k (2 /d)
(c)
Frequency (Hz)
700k
1.5M
1.0M
0.00
0.25
0.00
Transmission (au)
0.01
Transmission (au)
Figure 2: Comparison of the band structure (left side) with the experimental
transmission (right side). In panels (a), (b), (c) and (d) we present the results
for transducers of 250 kHz, 500 kHz, 1MHz and 1.5 MHz, respectively.
4
Transversal waves
Now we consider the transmission of transverse waves. The transverse velocities are ctal = 3100 m/s and ctep = 1160 m/s. In Fig. 3 we present in the
left side the band structure for longitudinal and transverse waves with dotted
and dashed lines, respectively. In the right side, we present the experimental
transmission obtained for a 500 kHz normal incidence shear wave transducer.
We observe that for the case of transverse waves, it exist a certain degree
of acoustical contamination from the longitudinal bands over the transverse
bands which can be observed in the transmission. It has been reported [19]
that for an homogeneous medium the contamination between the longitudinal
amplitude (Al ) over the transverse amplitude (At ) is on the order of −30dB.
The equivalence in the lineal scale can be calculated in the form
Al
−30dB = 20log
At
Al ≈ 0.03At
(2)
(3)
We have found that the contamination in the transmission can be much
more important for the multilayer structure than for an homogeneous medium
due to the existence of multiple reflections for the spurious longitudinal modes
which can cause constructive interference. On the other hand, it has been
Longitudinal and transverse elastic waves
0.0
kz (2 d)
23
0.5
700k
Frequency (Hz)
600k
500k
400k
300k
200k
0.00
0.27
Transmission (au)
Figure 3: Comparison of the band structure and the transverse wave transmission. In the left, we present the band structure for longitudinal and transverse
bands with dotted and dashed line respectively. In the right side we present
the transverse transmission, where the existence of the transverse bands is
verified, but also can be observed the contamination of longitudinal waves.
reported [20] that the absorption in the polymer is twice greater for transverse
waves than for longitudinal waves. These two effects contributed to observe
a transverse transmission that is the result of the combination of both bands,
transverse and longitudinal
In the right side of Fig. 3 we present with thick arrows the maximums
related with transverse waves. Conversely, we present with tiny arrows the
maximum related with longitudinal waves. We observe in the range from
250-350 kHz a clear sign of two transverse bands, but also we observe the
sign of a longitudinal band in the middle of these transverse bands. In the
range of 450-550 kHz, the transmission verify the existence of two transverse
bands. However, the existence of the band gap between is not well defined.
This probably is due that the transductor is centered in the 500 kHz and the
maximum power is at this frequency. Nevertheless, it exist a small attenuation
in the band gap just below the 500 kHz, which can be considered as the sign of
the band gap. Finally, in the range of 600 kHZ we observe a tiny manifestation
of a transverse band.
24
5
B. Manzanares-Martinez et al
Conclusions
We have presented experimental results of the transmission for longitudinal
and transverse waves in a finite one dimensional phononic crystal. We have
compared these results with the theoretical band structure. We have found
that the existence of bands and band gaps is easily demonstrated for longitudinal waves and it has been possible to verify the existence up to the 12th. band.
In contrast, we have found that the verification of transverse bands is difficult
due to the presence of contamination of longitudinal waves in the transverse
transductor. We have found that this contamination can be much more important in multilayers than in homogeneous media due to the existence of two
different phenomena. The first is related to the multiple reflections of the
spurious longitudinal modes that increases the longitudinal coherent transmission. The second is related to the bigger absorption that suffers the transverse
waves with respect to the longitudinal waves. We considered that these two
phenomena can be a limiting factor in study of phenomena related with the
propagation of transverse waves in phononic crystals.
ACKNOWLEDGEMENTS. We thank Gerardo Morales-Morales for
technical support. This work have been supported by the ”Division de CienciasURC” of the Universidad de Sonora, the grant URS11-PI07 of the Universidad
de Sonora and the project CB-2008/104151 of the ”Consejo Nacional de Ciencia y Tecnologia (CONACYT)”.
References
[1] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, Phys.
Rev. Lett. 71, 2022 (1993).
[2] E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[3] R. H. Olsson and I. El-Kady, Measurement Science and Technology 20,
012002 (2009).
[4] I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari- Rouhani, B.
Bonello, and V. Laude, Phys. Rev. B 82,174303 (2010).
[5] Z. Hou and B. M. Assouar, Journal of Physics D: Applied Physics 41,
095103 (2008).
[6] Z. G. Wang, S. H. Lee, C. K. Kim, C. M. Park, K. Nahm, and S. A.
Nikitov, Journal of Physics: Condensed Matter 20, 055209 (2008).
[7] Y. Cheng, X. J. Liu, and D. J. Wu, Journal of Applied Physics 109,
064904 (2011).
Longitudinal and transverse elastic waves
25
[8] B. Manzanares-Martinez, J. Sanchez-Dehesa, A. Hakans- son, F. Cervera,
and F. Ramos- Mendieta, Applied Physics Letters 85, 154 (2004).
[9] L.Y. Wu, M.L. Wu, and L.W. Chen, Smart Materials and Structures 18,
015011 (2009).
[10] Y. El Hassouani, E. H. El Boudouti, B. Djafari-Rouhani, and H. Aynaou,
Phys. Rev. B 78, 174306 (2008).
[11] Z. Xue-Feng, L. Sheng-Chun, X. Tao, W. Tie-Hai, and C. Jian-Chun,
Chinese Physics B 19, 044301 (2010).
[12] L. C. Parsons and G. T. Andrews, Applied Physics Letters 95, 241909
(2009).
[13] J. Manzanares-Martinez, R. Archuleta-Garcia, P. Castro-Garay, D.
Moctezuma-Enriquez, and E. Urrutia-Banuelos, Progress In Electromagnetics Research 111, 105 (2011).
[14] R. Archuleta-Garcia, D. Moctezuma-Enriquez, and J. ManzanaresMartinez, J. of Electromagn. Waves and Appl. 24, 351 (2010).
[15] X. Kang, W. Tan, Z. Wang, and H. Chen, Phys. Rev. A 79, 043832 (2009).
[16] T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, Phys. Rev. Lett.
101, 113902 (2008).
[17] Z. He, S. Peng, F. Cai, M. Ke, and Z. Liu, Phys. Rev. E 76, 056605
(2007).
[18] N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, O. Mauguin, L. Largeau, and A. Lematre, Phys. Rev. Lett. 104, 197402 (2010).
[19] http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/
EquipmentTrans/transducertypes.htm, cited july 14, 2011.
[20] Ultrasonic Methods for the Characterization of Complex Materials and
Material Systems: Polymers, Structured Polymers, Soft Tissue and Bone.
Charles Landais, Ph. D. Thesis (2011). University of Nebraska.
Received: August, 2011