Phononic crystals are artificial materials made of a periodic distribution of solid scatterers embedded into a solid host medium with different physical properties. An interesting case of phononic crystals, known as sonic crystals (SCs), appears when the solid scatterers are periodically embedded in a fluid medium. In SCs only longitudinal modes are allowed to propagate and both the theoretical and the experimental studies of the properties of the system are simplified without loss of generality. The most celebrated property of these systems is perhaps the existence of spectral band gaps. However, the periodicity of the system can also affect to the spatial dispersion, making possible the control of the diffraction inside these structures. In this work we study the main features of the spatial dispersion in SCs from a novel point of view taking into account the evanescent properties of the system, i.e., studying the complex spatial dispersion relations. The evanescent behavior of the propagation of waves in the angular band gaps are theoretically and experimentally observed in this work. Both the numerical predictions and the experimental results show the presence of angular band gaps in good agreement with the complex spatial dispersion relation. The results shown in this work are independent of the spatial scale of the structure, and in principle the fundamental role of the evanescent waves could be also expected in micro- or nanoscale phononic crystals.
Issue Section:
Research Papers
References
1.
Rayleigh
, J.
, 1887
, “Acoustical Observations
,” Philos. Mag.
, XXIV
, pp. 145
–159
.2.
Bloch
, F.
, 1928
, “Uber die Quantenmechanik der Electron in Kristallgittern
,” Z. Phys.
, 52
, pp. 555
–600
.3.
Brillouin
, L.
, 1953
, Wave Propagation in Periodic Structures
, 2nd ed. Dover
, New York
.4.
Yablonovitch
, E.
, 1987
, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics
,” Phys. Rev. Lett.
, 58
(20), pp. 2059
–2062
.10.1103/PhysRevLett.58.20595.
John
, S.
, 1987
, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices
,” Phys. Rev. Lett.
, 58
(23), pp. 2486
–2489
.10.1103/PhysRevLett.58.24866.
Ruffa
, A.
, 1992
, “Acoustic Wave Propagation Through Periodic Bubbly Liquids
,” J. Acoust. Soc. Am.
, 91
(1), pp. 1
–11
.10.1121/1.4027557.
Sigalas
, M.
, and Economou
, E.
, 1992
, “Elastic and Acoustic Wave Band Structure
,” J. Sound Vib.
, 158
(2), pp. 377
–382
.10.1016/0022-460X(92)90059-78.
Sigalas
, M.
, and Economou
, E.
, 1993
, “Band Structure of Elastic Waves in Two Dimensional Systems
,” Solid State Commun.
, 86
(3), pp. 141
–143
.10.1016/0038-1098(93)90888-T9.
Kushwaha
, M.
, Halevi
, P.
, Dobrzynski
, L.
, and Djafari-Rouhani
, B.
, 1993
, “Acoustic Band Structure of Periodic Elastic Composites
,” Phys. Rev. Lett.
, 71
(13
), pp. 2022
–2025
.10.1103/PhysRevLett.71.202210.
Kushwaha
, M.
, Halevi
, P.
, Martnez
, G.
, Dobrzynski
, L.
, and Djafari-Rouhani
, B.
, 1994
, “Theory of Acoustic Band Structure of Periodic Elastic Composites
,” Phys. Rev. B
49
(4
), pp. 2313
–2322
.10.1103/PhysRevB.49.231311.
Sigalas
, M.
, Economou
, E.
, and Kafesaki
, M.
, 1994
, “Spectral Gaps for Electromagnetic and Scalar Waves: Possible Explanation for Certain Differences
,” Phys. Rev. B
, 50
(5), pp. 3393
–3396
.10.1103/PhysRevB.50.339312.
Martínez-Sala
, R.
, Sancho
, J.
, Sánchez
, J. V.
, Gómez
, V.
, Llinares
, J.
, and Meseguer
, F.
, 1995
, “Sound Attenuation by Sculpture
,” Nature
, 378
, p. 241
.10.1038/378241a013.
Sánchez-Pérez
, J. V.
, Caballero
, D.
, Martínez-Sala
, R.
, Rubio
, C.
, Sánchez-Dehesa
, J.
, Meseguer
, F.
, Llinares
, J.
, and Gálvez
, F.
, 1998
, “Sound Attenuation by a Two-Dimensional Array of Rigid Cylinders
,” Phys. Rev. Lett.
, 80
(24
), pp. 5325
–5328
.10.1103/PhysRevLett.80.532514.
Kushwaha
, M.
, 1997
, “Stop-Bands for Periodic Metallic Rods: Sculptures That Can Filter the Noise
,” Appl. Phys. Lett.
, 70
(24), p. 3218
.10.1063/1.11913015.
Robertson
, W. M.
, and Rudy
J. F.
, III, 1998
, “Measurement of Acoustic Stop Bands in Two-Dimensional Periodic Scattering Arrays
,” J. Acoust. Soc. Am.
, 104
(2
), pp. 694
–699
.10.1121/1.42334416.
Khelif
, A.
, Choujaa
, A.
, Djafari-Rouhani
, B.
, Wilm
, M.
, Ballandras
, S.
, and Laude
, V.
, 2003
, “Trapping and Guiding of Acoustic Waves by Defect Modes in a Full-Band-Gap Ultrasonic Crystal
,” Phys. Rev. B
, 68
(21), p. 214301
.10.1103/PhysRevB.68.21430117.
Sánchez-Pérez
, J.
, Rubio
, C.
, Martínez-Sala
, R.
, Sánchez-Grandia
, R.
, and Gómez
, V.
, 2002
, “Acoustic Barriers Based on Periodic Arrays of Scatterers
,” Appl. Phys. Lett.
, 81
(27), p. 5240
.10.1063/1.153311218.
Romero-García
, V.
, and Sánchez-Pérez
, L. G.-R.
, 2011
, “Tunable Wideband Bandstop Acoustic Filter Based on Two-Dimensional Multiphysical Phenomena Periodic Systems
,” J. Appl. Phys.
, 110
(1), p. 014904
.10.1063/1.359988619.
Qiu
, C.
, Liu
, Z.
, Shi
, J.
, and Chan
, C. T.
, 2005
, “Directional Acoustic Source Based on the Resonant Cavity of Two-Dimensional Phononic Crystals
,” Appl. Phys. Lett.
, 86
(22), p. 224105
.10.1063/1.194264220.
Qiu
, C.
, and Liu
, Z.
, 2006
, “Acoustic Directional Radiation and Enhancement Caused by Band-Edge States of Two-Dimensional Phononic Crystals
,” Appl. Phys. Lett.
, 89
(6), p. 063106
.10.1063/1.233597521.
Sigalas
, M.
, 1998
, “Defect States of Acoustic Waves in a Two-Dimensional Lattice of Solid Cylinders
,” J. Appl. Phys.
, 84
(6), p. 3026
.10.1063/1.36845622.
Tanaka
, Y.
, Yano
, T.
, and Ichiro Tamura
, S.
, 2007
, “Surface Guided Waves in Two-Dimensional Phononic Crystals
,” Wave Motion
, 44
(6), pp. 501
–512
.10.1016/j.wavemoti.2007.02.00923.
Vasseur
, J. O.
, Deymier
, P. A.
, Djafari-Rouhani
, B.
, Pennec
, Y.
, and Hladky-Hennion
, A.-C.
, 2008
, “Absolute Forbidden Bands and Waveguiding in Two-Dimensional Phononic Crystal Plates
,” Phys. Rev. B
, 77
(8), p. 085415
.10.1103/PhysRevB.77.08541524.
Zhao
, Y.
, and Yuan
, L. B.
, 2009
, “Characteristics of Multi-Point Defect Modes in 2S Phononic Crystals
,” J. Phys. D: Appl. Phys.
, 42
(1), p. 015403
.10.1088/0022-3727/42/1/01540325.
Wu
, L.
, Chen
, L.
, and Liu
, C.
, 2009
, “Experimental Investigation of the Acoustic Pressure in Cavity of a Two-Dimensional Sonic Crystal
,” Physica B
, 404
(12–13), pp. 1766
–1770
.10.1016/j.physb.2009.02.02526.
Hussein
, M.
, 2009
, “Theory of Damped Bloch Waves in Elastic Media
,” Phys. Rev. B
, 80
(21), p. 212301
.10.1103/PhysRevB.80.21230127.
Romero-García
, V.
, Vasseur
, J. O.
, Hladky-Hennion
, A. C.
, Garcia-Raffi
, L. M.
, and Sánchez-Pérez
, J. V.
, 2011
, “Level Repulsion and Evanescent Waves in Sonic Crystals
,” Phys. Rev. B
, 84
(21), p. 212302
.10.1103/PhysRevB.84.21230228.
Farzbod
, F.
, and Leamy
, M.
, 2011
, “Analysis of Bloch's Method in Structures With Energy Dissipation
,” ASME J. Acoust. Vib.
, 133
(5), p. 051010
.10.1115/1.400394329.
Laude
, V.
, Moiseyenko
, R.
, Benchabane
, S.
, and Delercq
, N.
, 2011
, “Bloch Wave Deafness and Modal Conversion at a Phononic Cyrstal Boundary
,” AIP Adv.
, 1
(4), p. 041402
.10.1063/1.367582830.
Moiseyenko
, R.
, Herbison
, S.
, Delercq
, N.
, and Laude
, V.
, 2012
, “Phononic Crystal Diffraction Gratings
,” J. Appl. Phys.
, 111
(3), p. 034907
.10.1063/1.368211331.
Romero-García
, V.
, Vasseur
, J.
, Garcia-Raffi
, L.
, and Hladky-Hennion
, A.-C.
, 2012
, “Theoretical and Experimental Evidence of Level Repulsion States and Evanescent Modes in Sonic Crystal Stubbed Waveguides
,” New J. Phys.
, 14
, p. 023049
.10.1088/1367-2630/14/2/02304932.
Sánchez-Morcillo
, V.
, Staliunas
, K.
, Espinosa
, V.
, Pérez-Arjona
, I.
, Redondo
, J.
, and Soliveres
, E.
, 2009
, “Propagation of Sound Beams Behind Sonic Crystals
,” Phys. Rev. B
, 80
(13), p. 134303
.10.1103/PhysRevB.80.13430333.
Rakich
, P. T.
, Dahlem
, M. S.
, Tandon
, S.
, Ibanescu
, M.
, Soljai
, M.
, Petrich
, G. S.
, Joannopoulos
, J. D.
, Kolodziesjski
, L. A.
, and Ippen
, E. P.
, 2006
, “Achieving Centimeter-Scale Super-Collimation in Ultra Large Area Photonic Crystals
,” Nature Mater.
, 5
, pp. 93
–96
.10.1038/nmat156834.
Lu
, Z.
, Shi
, S.
, Murakowski
, A.
, Schneider
, G. J.
, Schuetz
, C. A.
, and Prather
, D. W.
, 2006
, “Experimental Demonstration of Self-Collimation Inside a Three-Dimensional Photonic Crystal
,” Phys. Rev. Lett.
, 96
(17), p. 173902
.10.1103/PhysRevLett.96.17390235.
Pérez-Arjona
, I.
, Sánchez-Morcillo
, V. J.
, Redondo
, J.
, Espinosa
, V.
, and Staliunas
, K.
, 2007
, “Theoretical Prediction of the Nondiffractive Propagation of Sonic Waves Through Periodic Acoustic Media
,” Phys. Rev. B
, 75
(1), p. 014304
.10.1103/PhysRevB.75.01430436.
Espinosa
, V.
, Sánchez-Morcillo
, V. J.
, Staliunas
, K.
, Pérez-Arjona
, I.
, and Redondo
, J.
, 2007
, “Subdiffractive Propagation of Ultrasound in Sonic Crystals
,” Phys. Rev. B
, 76
(14), p. 140302(R)
.10.1103/PhysRevB.76.14030237.
Luo
, C.
, Johnson
, S. G.
, Joannopoulos
, J. D.
, and Pendry
, J. B.
, 2002
, “All-Angle Negative Refraction Without Negative Effective Index
,” Phys. Rev. B
, 65
(20), p. 201104(R)
.10.1103/PhysRevB.65.20110438.
Luo
, C.
, Johnson
, S. G.
, Joannopoulos
, J. D.
, and Pendry
, J. B.
, 2003
, “Subwavelength Imaging in Photonic Crystals
,” Phys. Rev B
, 68
(4), p. 045115
.10.1103/PhysRevB.68.04511539.
Yang
, S.
, Page
, J. H.
, Liu
, Z.
, Cowan
, M. L.
, Chan
, C.
, and Sheng
, P.
, 2004
, “Focusing of Sound in a 3D Phononic Crystal
,” Phys. Rev. Lett.
, 93
(2
), p. 024301
.10.1103/PhysRevLett.93.02430140.
Ke
, M.
, Liu
, Z.
, Qiu
, C.
, Wang
, W.
, Shi
, J.
, Wen
, W.
, and Sheng
, P.
, 2005
, “Negative-Refraction Imaging With Two-Dimensional Phononic Crystals
,” Phys. Rev. B
, 72
(6), p. 064306
.10.1103/PhysRevB.72.06430641.
Feng
, L.
, Liu
, X.
, Chen
, Y.
, Huang
, Z.
, Mao
, Y.
, Chen
, Y.
, Zi
, J.
, and Zhu
, Y.
, 2005
, “Negative Refraction of Acoustic Waves in Two-Dimensional Sonic Crystals
,” Phys. Rev. B
, 72
(3), p. 033108
.10.1103/PhysRevB.72.03310842.
Romero-García
, V.
, Sánchez-Pérez
, J.
, and Garcia-Raffi
, L.
, 2010
, “Evanescent Modes in Sonic Crystals: Complex Dispersion Relation and Supercell Approximation
,” J. Appl. Phys.
, 108
(4), p. 044907
.10.1063/1.346698843.
Laude
, V.
, Achaoui
, Y.
, Benchabane
, S.
, and Khelif
, A.
, 2009
, “Evanescent Bloch Waves and the Complex Band Structure of Phononic Crystals
,” Phys. Rev. B
, 80
(9), p. 092301
.10.1103/PhysRevB.80.09230144.
Romero-García
, V.
, Sánchez-Pérez
, J.
, and Garcia-Raffi
, L.
, 2010
, “Propagating and Evanescent Properties of Double-Point Defects in Sonic Crystals
,” New. J. Phys.
, 12
, p. 083024
.10.1088/1367-2630/12/8/08302445.
Romero-García
, V.
, Sánchez-Pérez
, J.
, Neira Ibáñez
, S. C.
, and Garcia-Raffi
, L.
, 2010
, “Evidences of Evanescent Bloch Waves in Phononic Crystals
,” Appl. Phys. Lett.
, 96
(12), p. 124102
.10.1063/1.336773946.
Romero-García
, V.
, Garcia-Raffi
, L. M.
, and Sánchez-Pérez
, J. V.
, 2011
, “Evanescent Waves and Deaf Bands in Sonic Crystals
,” AIP Adv.
, 1
(4), p. 041601
.10.1063/1.367580147.
Romero-García
, V.
, 2010
, “On the Control of Propagating Acoustic Waves in Sonic Crystals: Analytical, Numerical and Optimization Techniques
,” Ph.D. thesis, Instituto de Cienca de Materiales de Madrid (CSIC), Universitat Politècnica de València, València, Spain.48.
Li
, J.
, and Chan
, C.
, 2004
, “Double-Negative Acoustic Metamaterial
,” Phys. Rev. E
, 70
(5), p. 055602
.10.1103/PhysRevE.70.05560249.
Guenneau
, S.
, Movchan
, A.
, Ptursson
, G.
, and Ramakrishna
, S. A.
, 2007
, “Acoustic Metamaterial for Sound Focusing and Confinement
,” New J. Phys.
, 9
(11), p. 399
.10.1088/1367-2630/9/11/39950.
Farhat
, M.
, Guenneau
, S.
, Enoch
, S.
, Tayeb
, G.
, Movchan
, A. B.
, and Movchan
, N. V.
, 2008
, “Analytical and Numerical Analysis of Lensing Effect for Linear Surface Water Waves Through a Square Array of Nearly Touching Rigid Square Cylinders
,” Phys. Rev. E
, 77
(4), p. 046308
.10.1103/PhysRevE.77.04630851.
Hsue
, Y.
, Freeman
, A.
, and Gu
, B.
, 2005
, “Extended Plane-Wave Expansion Method in Three-Dimensional Anisotropic Photonic Crystals
,” Phys. Rev B
, 72
(19), p. 195118
.10.1103/PhysRevB.72.19511852.
Martin
, P.
, 2006
, Multiple Scattering. Interaction of Time-Harmonic Waves With N Obstacles
, Cambridge University Press
, Cambridge, UK
.53.
Chen
, Y. Y.
, and Ye
, Z.
, 2001
, “Theoretical Analysis of Acoustic Stop Bands in Two-Dimensional Periodic Scattering Arrays
,” Phys. Rev. E
, 64
(3), p. 036616
.10.1103/PhysRevE.64.03661654.
McGaughey
, A. J. H.
, Hussein
, M. I.
, Landry
, E. S.
, Kaviany
, M.
, and Hulbert
, G. M.
, 2006
, “Phonon Band Structure and Thermal Transport Correlation in a Layered Diatomic Crystal
,” Phys. Rev. B
, 74
(10), p. 104304
.10.1103/PhysRevB.74.10430455.
Landry
, E. S.
, Hussein
, M. I.
, and McGaughey
, A. J. H.
, 2008
, “Complex Superlattice Unit Cell Designs for Reduced Thermal Conductivity
,” Phys. Rev. B
, 77
(18), p. 184302
.10.1103/PhysRevB.77.18430256.
Hopkins
, P. E.
, Reinke
, C. M.
, Su
, M. F.
, III, Shaner
, E. A.
, Leseman
, Z. C.
, Serrano
, J. R.
, Phinney
, L. M.
, and El-Kady
, I.
, 2011
, “Reduction in the Thermal Conductivity of Single Crystalline Silicon by Phononic Crystal Patterning
,” Nano Lett.
, 11
(1), pp. 107
–112
.10.1021/nl102918q57.
Davis
, B. L.
, and Hussein
, M. I.
, 2011
, “Thermal Characterization of Nanoscale Phononic Crystals Using Supercell Lattice Dynamics
,” AIP Adv.
, 1
(4), p. 041701
.10.1063/1.367579858.
Gorishnyy
, T.
, Ullal
, C. K.
, Maldovan
, M.
, Fytas
, G.
, and Thomas
, E. L.
, 2005
, “Hypersonic Phononic Crystals
,” Phys. Rev. Lett.
, 94
(11), p. 115501
.10.1103/PhysRevLett.94.115501Copyright © 2013 by ASME
You do not currently have access to this content.
