“Homogenization of photonic and phononic crystals” F. Pérez Rodríguez Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, México E-mail: [email protected]International Jubilee Seminar “Current Problems in Solid State Physics” November 15-19, 2011, Kharkov, Ukraine
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“Homogenization of photonic and phononic crystals” F. Pérez Rodríguez
International Jubilee Seminar “Current Problems in Solid State Physics” November 15-19, 2011, Kharkov, Ukraine. “Homogenization of photonic and phononic crystals” F. Pérez Rodríguez Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, M éxico - PowerPoint PPT Presentation
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“Homogenization of photonic and phononic crystals”
F. Pérez Rodríguez
Instituto de Física, Benemérita Universidad Autónoma de Puebla,Apdo. Post. J-48, Puebla, Pue. 72570, México
•Arbitrary dielectric, metallic, magnetic, and chiral inclusions.
•Arbitrary Bravais lattice.
•Inclusions in neighboring cells can be isolated or in contact.
Material characterizationTensors of the bianisotropic response
Particular cases: magnetodielectric and metallomagnetic photonic crystals with isotropic inclusions
)(
)(
)(
)(
)(
)(
rh
re
Ir0
0Ir
rb
rd��
��
)(
)(
)()(
)()(
)(
)(
rh
re
rμr
rξrε
rb
rd��
��
)(
)()(
rh
rerv
)()()( rvrArv0I
I0
i��
��Maxwell’s Equations at micro-level
)()(
)()(
rμr
rξrεA ��
��
Homogenization of Photonic Crystals
V. Cerdán-Ramírez, B. Zenteno-Mateo, M. P. Sampedro, M. A. Palomino-Ovando, B. Flores-Desirena, and F. Pérez-Rodríguez, J. Appl. Phys. 106, 103520 (2009).
G
rGGArA ie)()(
A photonic crystal being periodic by definition:
0GvGGkDG
)'()',;('
)'((
()',;( ', GGA
0IG)k
IG)k0GGkD GG
��
��
Master equation
Macroscopic fields
effeff
effeffeff μ
ξεA ��
��
Effective parameters
)()(
)()(
rμr
rξrεA ��
��
Homogenization
11
),;(1
000kD
μ
ξεA
effeff
effeffeff ��
��
Cubic lattice of small spheres
I0
0I
μ
ξεA
��
��
��
��
baab
baabb
baab
baabb
efef
efefef
ff
ff
2
22
222
Maxwell Garnett
Cubic and Orthorhombic PCs
Cubic and Orthorhombic PCs
Cubic lattices
Cubic lattices
Metallic wires
0/'' zz
0.0 0.5 1.0 1.5 2.0-10
-8
-6
-4
-2
0
2
105, 106104
p=103
Re
a /c
0.0 0.5 1.0 1.5 2.00
2
4
6
8
10
106
105
104
p=103Im
a /c
f = 0.001
r/a = 0.017
p = cμ0 a σ
0/' zz
z
Pendry´s formula
Magnetic wires
High-permeability metals and alloys
Magnetic properties of various grades of iron
zz'
High-permeability magnetic wiresz
1000+10i
0 0.1 0.2
Left-handed metamaterial
xzy
0,0 yyzz
Left-handed metamaterial
Magnetometallic PC
300+5i 1000+10i
Rytov (1956)
Effective plasma frequency for metal-dielectric superlattices
Effective permittivity
Metal-dielectric superlattice
B. Zenteno-Mateo, V. Cerdán-Ramírez, B. Flores-Desirena, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, Progress in Electromagnetics Research Letters (PIER Lett.) 22, 165-174 (2011)
Xu et al (2005)
f=0.5/10.5
PIER Lett. (2011)
Al-glass
Al-glass
Al-glass
f=0.5/100.5
IGGiGkGkIGkGGNGG
��� )'(ˆ)])(()|[(|)',( 0'
20
2 k
J.A. Reyes-Avendaño, U. Algredo-Badillo, P. Halevi, and F Pérez-Rodríguez, New J. Phys. 13 073041 (2011).
Material characterization(conductivity)
Nonlocal effective conductivity dyadic:
Nonlocal dielectric response
Magneto-dielectric response
Bianisotropic response
Expansion in small wavevectors (ka<< 1):
3D crosses of continous wires
New J. Phys. (2011)
3D crosses of cut wires
3D crosses of cut wires
Continuous wires
Cut wires
Cut wires
3D crosses of asymmetrically-cut wires
“Elastic metamaterials”
F. Pérez RodríguezInstituto de Física, Benemérita Universidad Autónoma de Puebla,
Mexico
International Jubilee Seminar “Current Problems in Solid State Physics”
dedicated to the memory of Associate member of National Academy of Sciences of Ukraine
E. A. Kaner and 55th anniversary of discovery of Azbel-Kaner cyclotron resonance
November 16-18, 2011, Kharkov, Ukraine
Plan
1. Phononic crystals
2. Homogenization theory
3. Comparison with other approaches
4. Elastic metamaterials
Phononic crystals
(r), Cl(r), Ct(r)
Wave equation:
G
rGieGr
·)()( G
rGil eGCrC
·211 )()(
G
rGit eGCrC
·244 )()(
Photonic crystalPhotonic metamaterial
Phononic crystalPhononic metamaterial
ef
ef
eff, Ct,eff Cl,eff
New J. Phys. 13, 073041 (2011)
J. Appl. Phys 106, 103520 (2009)
Phononic metamaterials
cnk /||
/n
Similarity with photonic metamaterials
1. Poynting vector and wave vector are oposite if the mass density is negative
2. The refraction index is real (negative) if the density and elastic (bulk) modulus are both negative
In the photonic case:
Phononic metamaterials
¿How can one obtain a negative mass?
Resonant sonic materials
Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, P. Sheng, Science, 2000.
Z. Yang, J. Mei, M. Yang, N. H. Chan, P. Sheng, PRL, 2008
Membrane-Type Acoustic Metamaterial with Negative Dynamic Mass
afmD /
H. Chena, C. T. Chan, APL, 2007
Acoustic cloacking
Homogenization of phononic crystals
lkijklij
ijji
urC
ur
)(
)(2
0
0
0
00
00
00
12
13
23
3
2
1
63
6
5
4
3
2
1
3
2
1
u
u
u
V
636
6332
0
0
I
Is
G
rGiK
trKi eGVetrV
·)·( )(),(
Bloch wave:
0
··· )()0(G
rGiK
rKiK
rKi eGVeVe
Master equation:
0 ')G(V')G,G;k(D'G
G')(GAωiδGkK
GkK')G,G;k(D sG,G'T
663
633
0)(
)(0
'0
0''
6636
633
GGS
IGGGGA
Equations at macroscopic level
Effective parameters
111 )0,0;0,( kDiA seff
Local response:
Nonlocal response:
663
6331
111
0)(
)(0
)0,0;,(
Ts
seff
kK
kKi
kDiA
eff
effeff S
A36
63
0
0�Homogenization
)(0
0)()(
36
63
rS
IrrA
�
0,0 0,2 0,4 0,6 0,8 1,02,00E+010
3,00E+010
4,00E+010
5,00E+010
6,00E+010
7,00E+010
8,00E+010
9,00E+010
1,00E+011
1,10E+011
1,20E+011
1,30E+011
1,40E+011
1,50E+011
1,60E+011
1,70E+011
Pa
f
C33 C22 C11 C23 C12 C13 C66 C55 C44
Si/Al 1D phononic crystals
0,0 0,2 0,4 0,6 0,8 1,02300
2350
2400
2450
2500
2550
2600
2650
2700
2750
kg /
m3
f
XX
YY
ZZ
Comparison with numerical results:José A. Otero Hernández1, Reinaldo Rodríguez2, Julián Bravo2
1 Instituto de Cibernética, Matemática y Física. (ICIMAF), Cuba2 Facultad de Matemática y Computación, UH, Cuba.