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, México
E-mail: fperez@ifuap.buap.mx
Plan
1. Metamateriales fotónicos
2. Metamateriales fonónicos
 ef
 ef
Photonic
crystal
Photonic
metamaterial
Refraction index
2
n ef
= ef ef
n ef = ef ef
 ef ef , if Re ef  0 , Re  ef  0
n ef  
 ef ef , if Re ef  0 , Re  ef  0
Photonic metamaterial
Pendry and Smith, Phys.Today (2004)
Poynting and wave vectors
Positive- index or right-handed material.
Negative-index or left- handed material.
Refracción negativa
fuente
 p ( )  0
kp
 p ( )  0
Sp
p
k‫װ‬
n
kn
Sn
n p ( )  0
 n ( )  0
 n ( )  0
nn ( )  0
Simulation of refraction
Pendry and Smith, Phys.Today (2004).
Observation of negative refraction
Shelby, Smith and Schultz, Science (2001)
J. Valentine, S. Zhang, T. Zentgraf, et al,
Nature, 2008
n    
E. Plum, et al (2009)
Focusing with ordinary and Veselago lenses
Pendry and Smith, Phys.Today (2004).
How to “make” the PC uniform?
Conventional approach:
(Bloch) wavelength >> lattice constant (period)
 B = 2 k >> a
Homogenization or mean-field theory
Rapid oscillations of fields are smoothed out:
Er  , B r 
 eikr
Theory is very general:
•Arbitrary dielectric, metallic, magnetic, and chiral
inclusions.
•Arbitrary Bravais lattice.
•Inclusions in neighboring cells can be isolated or
in contact.
Material characterization
Tensors of the bianisotropic response


d(r )  ε (r ) ξ (r )   e(r ) 
 
b(r )   


  (r ) μ(r ) h(r )
Particular cases: magnetodielectric and metallomagnetic photonic crystals with
isotropic inclusions


0   e(r) 
d(r)  (r) I
  
b(r)   0

 (r) I  h(r)

 
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).
Maxwell’s Equations at micro-level


  I
 0
   v(r)  iA(r )  v(r)


0 
   I
 e(r) 
v(r)  

h(r)


 ε (r ) ξ (r ) 
A  
 
 (r ) μ (r ) 
A photonic crystal being periodic by definition:
A(r)   A(G)e
G
iGr
Master equation
 D(k; G, G' )  v(G' )  0
G'


0
(k  G) I 

D(k; G, G ' )  


 G ,G '   A(G  G ' )
0
 (k  G) I

Macroscopic fields
Effective parameters
Aeff

εeff
 
 eff


 ε (r ) ξ (r ) 
A  
 
 (r ) μ (r ) 

1
ξ eff  1  1

    D (k  0;0,0)
μ eff   

Homogenization
Aeff

εeff
 
 eff

ξ eff 
 
μ eff 
Cubic lattice of small spheres
Maxwell Garnett

 ε ef
A ef   
 ef
  2 b   a  2 f  a   b   

 I
 b 

ξ ef    2 b   a  f  a   b  
 

μ ef 
0





 2b   a  2 f  a  b    
 I 
b 


2




f



b
a
a
b  


0
Cubic and Orthorhombic PCs
Cubic and Orthorhombic PCs
Cubic lattices
Cubic lattices
Metallic wires
z
f = 0.001
r/a = 0.017
p = cμ0 a σ
 ' 'zz /  0
 ' zz /  0
10
2
0
8
-2
6
Im
Re
3
p=10
-4
4
3
-6
4
10
5
p=10
5
6
10 , 10
10
2
-8
4
6
10
10
0
-10
0.0
0.5
1.0
a /c
1.5
2.0
0.0
0.5
1.0
a /c
1.5
2.0
Pendry´s formula
Magnetic wires
High-permeability metals and alloys
Magnetic properties of various grades of iron
High-permeability magnetic wires
z
1000+10i
 ' zz
0
0.1
0.2
Left-handed metamaterial
 zz  0,
 yy  0
y
z
x
Left-handed metamaterial
Magnetometallic PC
300+5i
1000+10i
Effective plasma frequency for metal-dielectric superlattices
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)
Effective permittivity
Rytov (1956)
Metal-dielectric superlattice
Xu et al (2005)
Al-glass
f=0.5/10.5
PIER Lett. (2011)
Al-glass
Al-glass
f=0.5/100.5
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:
  
  
  2 2     
N(G, G' )  [(| k  G | k0 )I  (k  G)(k  G)]G G '  i0ˆ (G  G' )I
Nonlocal dielectric response
Expansion in small wave
vectors (ka<< 1):
Magneto-dielectric response
Bianisotropic response
3D crosses of continous wires
3D crosses of cut wires
New J. Phys. (2011)
3D crosses of cut wires
Continuous wires
Cut wires
Cut wires
3D crosses of asymmetrically-cut wires
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
“Elastic metamaterials”
F. Pérez Rodríguez
Instituto de Física, Benemérita Universidad Autónoma de Puebla,
Mexico
Plan
1. Phononic crystals
2. Homogenization theory
3. Comparison with other approaches
4. Elastic metamaterials
Phononic crystals
(r), Cl(r), Ct(r)
Wave equation:
 iG ·r

 (r )    (G)e

G
 iG ·r

2
C11 (r )  Cl   (G)e

G
 iG ·r

2
C44 (r )  Ct   (G)e

G
Photonic crystal
Photonic metamaterial
 ef
 ef
J. Appl. Phys 106,
103520 (2009)
New J. Phys. 13,
073041 (2011)
Phononic crystal
Phononic metamaterial
eff, Ct,eff
Cl,eff
Phononic metamaterials
Similarity with photonic metamaterials
k | n |  / c
n   /
In the photonic case:
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
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.
Membrane-Type Acoustic Metamaterial with Negative Dynamic Mass
mD  f  /  a 
Z. Yang, J. Mei, M. Yang, N. H. Chan, P. Sheng, PRL, 2008
Acoustic cloacking
H. Chena, C. T. Chan, APL, 2007
Homogenization of phononic crystals

   (r )ui   j ij

 ij  Cijkl (r ) k ul
2
 36
 1 0

  0 2
0
0

0
3
0 3
3  2
0
1
0
2 

1 
0 
  2 I 3
s  
 0 63
036 

I6 
 u1 
 
 u2 
u 
 3

  1
V   2 
 
 3 
 4 
 
 5 
 
 6
Bloch wave:





i ( K ·r t )
iG·r

V (r , t )  e
 VK (G)e
G
e
 

  iG ·r
iK ·r
VK (0)  e VK (G)e
 
iK ·r

G 0
Master equation:
    
 D(k ;G,G') V(G')  0
G'
 
K 36 (k  G )
 δG,G'  i ωs A(G  G')
06

  

03
D(k ;G,G')   
  T
 K 36 (k  G )


 

A G  G'  





 
G  G' I 3
063

036
  
S 66 G  G ' 


Equations at macroscopic level
Effective parameters
Nonlocal response:

1
Aeff  is D (, k ;0,0) 
1
1
03

is 
 T
K36 (k ) 
1

K36 (k )

06 
Local response:

1
Aeff  is D (, k  0;0,0)
1
1
 
  (r ) I
A (r )  
 063
Homogenization
036 
 
S ( r )

  eff 036 
Aeff  

063 Seff 
Si/Al 1D phononic crystals
XX
2750
2700
YY
1,70E+011
ZZ
1,60E+011
C33
C22
C11
C23
C12
C13
C66
C55
C44
1,50E+011
2650
1,40E+011
1,30E+011
1,20E+011
1,10E+011
2550
Pa
kg / m
3
2600
2500
1,00E+011
9,00E+010
8,00E+010
2450
7,00E+010
6,00E+010
2400
5,00E+010
4,00E+010
2350
3,00E+010
2300
2,00E+010
0,0
0,2
0,4
0,6
f
0,8
1,0
0,0
0,2
0,4
0,6
0,8
f
Comparison with numerical results:
José A. Otero Hernández1, Reinaldo Rodríguez2, Julián Bravo2
1 Instituto de Cibernética, Matemática y Física. (ICIMAF), Cuba
2 Facultad de Matemática y Computación, UH, Cuba.
1,0
Si/Al 2D phononic crystals
XX
2750
2700
YY
1,70E+011
ZZ
1,60E+011
C11
C12
C13
C33
C44
C66
1,50E+011
2650
1,40E+011
1,30E+011
1,20E+011
1,10E+011
2550
Pa
kg / m
3
2600
2500
1,00E+011
9,00E+010
8,00E+010
2450
7,00E+010
6,00E+010
2400
5,00E+010
4,00E+010
2350
3,00E+010
2300
2,00E+010
0,0
0,2
0,4
0,6
f
0,8
1,0
0,0
0,2
0,4
0,6
f
0,8
1,0
2D sonic crystal, solid in water (Al in water)
XX
YY
2400
ZZ
2200
2000
Kg / m
3
1800
1600
1400
1200
1000
0,0
0,2
0,4
0,6
0,8
f
2.4
Teoría Convencional
Cuadrada
Hexagonal
2.2
Ceff / Cb
effb
2.0
1.8
1.6
1.4
1.2
1.0
0.0
0.1
0.2
0.3
r/a
0.4
0.5
1.90
1.85
1.80
1.75
1.70
1.65
1.60
1.55
1.50
1.45
1.40
1.35
1.30
1.25
1.20
1.15
1.10
1.05
1.00
0.95
0.90
Cuadrada
Hexagonal
Teoría Convencional
0.0
0.1
0.2
0.3
r/a
0.4
0.5
Comparison with: D. Torrent, J. Sánchez-Dehesa, NJP (2008):
2.6
Hexagonal (Dr. Dehesa)
Square (Dr. Dehesa)
Square (T. Local)
Hexagonal (T. Local)
2.4
2.2
effb
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.0
0.1
0.2
0.3
R/a
0.4
0.5
Metamaterial response
Al/Rubber 1D phononic crystal
Transverse modes
1/ s)
60000
40000
20000
0
-300 -250 -200 -150 -100 -50
0
50
Kz (1/m)
100 150 200 250 300
Acoustic branch
Local
Nonlocal
Nonlocal
Local
6000
KZ T. Local
rad / seg
5000
KZ=/(C44,EF/EF)
KZ Exacto
4000
3000
2000
1000
0
0
50
100
150
KZ
200
250
300
First “optical” band
2000
EF (KZ-finito)
1000
EF (Teoría Local)
C44EF (KZ-finito)
-5.00E+007
C44EF (Teoría Local)
Local
0
-1.00E+008
Local
-2000
-3000
Pa
kg / m
3
-1000
Nonlocal
-1.50E+008
Nonlocal
-4000
-5000
-2.00E+008
-6000
-7000
-2.50E+008
55600 55800 56000
56200 56400 56600 56800 57000 57200
55600 55800 56000 56200 56400 56600 56800 57000 57200
rad / seg
rad / seg
57000
KZ T. Local
56800
Kz=/(C44,Ef+i)/(Ef-i)
KZ=/(C44,EF/EF)
56800
KZ Exacto
56600
rad.seg
rad / seg
56600
56400
56400
56200
56200
56000
56000
0
50
100
150
KZ
200
250
300
55800
-350
-300
-250
-200
-150
Kz
-100
-50
0
50
¡Gracias!