“Homogenization of photonic and phononic crystals” F. Pérez Rodríguez
description
Transcript of “Homogenization of photonic and phononic crystals” F. Pérez Rodríguez
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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
E-mail: [email protected]
International Jubilee Seminar “Current Problems in Solid State Physics”
November 15-19, 2011, Kharkov, Ukraine
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Plan
1. Metamateriales fotónicos
2. Metamateriales fonónicos
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Photonic crystal
Photonic metamaterial
ef
ef
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ef ef ef ef
ef
ef ef ef ef
, if Re 0 , Re 0 n
, if Re 0 , Re 0
Refraction index
2efn = ef ef
efn = ef ef
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Pendry and Smith, Phys.Today (2004)
Photonic metamaterial
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Poynting and wave vectors
Positive- index or right-handed material.
Negative-index or left- handed material.
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kp
Sp
kn
Sn
kװ
fuente
Refracción negativa
0)(
0)(
0)(
p
p
p
n
0)(
0)(
0)(
n
n
n
n
p
n
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Simulation of refraction
Pendry and Smith, Phys.Today (2004).
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Shelby, Smith and Schultz, Science (2001)
Observation of negative refraction
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J. Valentine, S. Zhang, T. Zentgraf, et al, Nature, 2008
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n
E. Plum, et al (2009)
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Pendry and Smith, Phys.Today (2004).
Focusing with ordinary and Veselago lenses
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B = 2 k >> a
i , e k rE r B r
How to “make” the PC uniform?
Homogenization or mean-field theory
Rapid oscillations of fields are smoothed out:
Conventional approach: (Bloch) wavelength >> lattice constant (period)
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Theory is very general:
•Arbitrary dielectric, metallic, magnetic, and chiral inclusions.
•Arbitrary Bravais lattice.
•Inclusions in neighboring cells can be isolated or in contact.
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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��
��
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)(
)()(
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).
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G
rGGArA ie)()(
A photonic crystal being periodic by definition:
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0GvGGkDG
)'()',;('
)'((
()',;( ', GGA
0IG)k
IG)k0GGkD GG
��
��
Master equation
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Macroscopic fields
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effeff
effeffeff μ
ξεA ��
��
Effective parameters
)()(
)()(
rμr
rξrεA ��
��
Homogenization
11
),;(1
000kD
μ
ξεA
effeff
effeffeff ��
��
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Cubic lattice of small spheres
I0
0I
μ
ξεA
��
��
��
��
baab
baabb
baab
baabb
efef
efefef
ff
ff
2
22
222
Maxwell Garnett
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Cubic and Orthorhombic PCs
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Cubic and Orthorhombic PCs
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Cubic lattices
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Cubic lattices
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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
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Pendry´s formula
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Magnetic wires
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High-permeability metals and alloys
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Magnetic properties of various grades of iron
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zz'
High-permeability magnetic wiresz
1000+10i
0 0.1 0.2
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Left-handed metamaterial
xzy
0,0 yyzz
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Left-handed metamaterial
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Magnetometallic PC
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300+5i 1000+10i
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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)
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Xu et al (2005)
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f=0.5/10.5
PIER Lett. (2011)
Al-glass
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Al-glass
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Al-glass
f=0.5/100.5
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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:
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Nonlocal dielectric response
Magneto-dielectric response
Bianisotropic response
Expansion in small wavevectors (ka<< 1):
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3D crosses of continous wires
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New J. Phys. (2011)
3D crosses of cut wires
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3D crosses of cut wires
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Continuous wires
Cut wires
Cut wires
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3D crosses of asymmetrically-cut wires
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“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
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Plan
1. Phononic crystals
2. Homogenization theory
3. Comparison with other approaches
4. Elastic metamaterials
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Phononic crystals
(r), Cl(r), Ct(r)
Wave equation:
G
rGieGr
·)()( G
rGil eGCrC
·211 )()(
G
rGit eGCrC
·244 )()(
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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)
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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:
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Phononic metamaterials
¿How can one obtain a negative mass?
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Resonant sonic materials
Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, P. Sheng, Science, 2000.
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Z. Yang, J. Mei, M. Yang, N. H. Chan, P. Sheng, PRL, 2008
Membrane-Type Acoustic Metamaterial with Negative Dynamic Mass
afmD /
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H. Chena, C. T. Chan, APL, 2007
Acoustic cloacking
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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
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G
rGiK
trKi eGVetrV
·)·( )(),(
Bloch wave:
0
··· )()0(G
rGiK
rKiK
rKi eGVeVe
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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
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Equations at macroscopic level
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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
�
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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.
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Si/Al 2D phononic crystals
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
C11 C12 C13 C33 C44 C66
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
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2D sonic crystal, solid in water (Al in water)
0,0 0,2 0,4 0,6 0,8
1000
1200
1400
1600
1800
2000
2200
2400
Kg
/ m
3
f
XX
YY
ZZ
0.0 0.1 0.2 0.3 0.4 0.50.900.951.001.051.101.151.201.251.301.351.401.451.501.551.601.651.701.751.801.851.90
Cuadrada Hexagonal
Cef
f / C
b
r/a
Teoría Convencional
0.0 0.1 0.2 0.3 0.4 0.5
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
eff b
r/a
Cuadrada Hexagonal
Teoría Convencional
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0.0 0.1 0.2 0.3 0.4 0.50.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
eff
b
Hexagonal (Dr. Dehesa) Square (Dr. Dehesa) Square (T. Local) Hexagonal (T. Local)
R/a
Comparison with: D. Torrent, J. Sánchez-Dehesa, NJP (2008):
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Metamaterial responseAl/Rubber 1D phononic crystal
Transverse modes
-300 -250 -200 -150 -100 -50 0 50 100 150 200 250 3000
20000
40000
60000
1
/ s)
Kz (1/m)
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Acoustic branch
0 50 100 150 200 250 3000
1000
2000
3000
4000
5000
6000
rad
/ seg
KZ
KZ T. Local
KZ=/(C
44,EF/
EF)
KZ Exacto
Local
NonlocalNonlocal
Local
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First “optical” band
55600 55800 56000 56200 56400 56600 56800 57000 57200
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
kg /
m3
rad / seg
EF
(KZ-finito)
EF
(Teoría Local)
55600 55800 56000 56200 56400 56600 56800 57000 57200-2.50E+008
-2.00E+008
-1.50E+008
-1.00E+008
-5.00E+007 C44EF
(KZ-finito)
C44EF
(Teoría Local)
Pa
rad / seg
0 50 100 150 200 250 300
56000
56200
56400
56600
56800 KZ T. Local
KZ=/(C
44,EF/
EF)
KZ Exacto
rad
/ seg
KZ
-350 -300 -250 -200 -150 -100 -50 0 5055800
56000
56200
56400
56600
56800
57000
Kz=/(C
44,Ef+i)/(
Ef-i)
rad.
seg
Kz
Nonlocal
Local
Local
Nonlocal
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¡Gracias!