“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

International Jubilee Seminar “Current Problems in Solid State Physics”

November 15-19, 2011, Kharkov, Ukraine

Plan

1. Metamateriales fotónicos

2. Metamateriales fonónicos

Photonic crystal

Photonic metamaterial

ef

ef

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

Pendry and Smith, Phys.Today (2004)

Photonic metamaterial

Poynting and wave vectors

Positive- index or right-handed material.

Negative-index or left- handed material.

kp

Sp

kn

Sn

kװ

fuente

Refracción negativa

0)(

0)(

0)(

p

p

p

n

0)(

0)(

0)(

n

n

n

n

p

n

Simulation of refraction

Pendry and Smith, Phys.Today (2004).

Shelby, Smith and Schultz, Science (2001)

Observation of negative refraction

J. Valentine, S. Zhang, T. Zentgraf, et al, Nature, 2008

n

E. Plum, et al (2009)

Pendry and Smith, Phys.Today (2004).

Focusing with ordinary and Veselago lenses

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)

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 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.

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

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

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):

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)

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

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

¡Gracias!