“Homogenization of photonic and phononic crystals” F. Pérez Rodríguez Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue , México 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

Refraction index

Pendry and Smith, Phys.Today (2004) Photonic metamaterial

Poynting and wave vectors Positive- index or right-handed material. Negative-index or left- handed material.

kpkp SpSp knkn SnSn kװkװ fuente Refracción negativa

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

E. Plum, et al (2009)

Pendry and Smith, Phys.Today (2004). Focusing with ordinary and Veselago lenses

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 characterization Tensors of the bianisotropic response Particular cases: magnetodielectric and metallomagnetic photonic crystals with isotropic inclusions

Maxwell’s Equations at micro-level 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, (2009).

A photonic crystal being periodic by definition:

Master equation

Macroscopic fields

Effective parameters Homogenization

Cubic lattice of small spheres Maxwell Garnett

Cubic and Orthorhombic PCs

Cubic lattices

Metallic wires f = r/ a = p = c μ 0 a σ z

Pendry´s formula

Magnetic wires

High-permeability metals and alloys

Magnetic properties of various grades of iron

High-permeability magnetic wires z i

Left-handed metamaterial x zy

Magnetometallic PC

300+5i i

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, (2011)

Xu et al (2005)

f=0.5/10.5 PIER Lett. (2011) Al-glass

f=0.5/100.5

J.A. Reyes-Avendaño, U. Algredo-Badillo, P. Halevi, and F Pérez-Rodríguez, New J. Phys (2011). Material characterization (conductivity) Nonlocal effective conductivity dyadic:

Nonlocal dielectric response Magneto-dielectric response Bianisotropic response Expansion in small wave vectors (ka<< 1):

3D crosses of continous wires

New J. Phys. (2011) 3D crosses of cut wires

Continuous wires Cut wires

3D crosses of asymmetrically-cut wires

“Elastic metamaterials” F. Pérez Rodríguez Instituto 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 55 th 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), C l (r), C t (r) Wave equation:

Photonic crystal Photonic metamaterial Phononic crystal Phononic metamaterial  eff, C t,eff C l,eff New J. Phys. 13, (2011) J. Appl. Phys 106, (2009)

Phononic metamaterials 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

H. Chena, C. T. Chan, APL, 2007 Acoustic cloacking

Homogenization of phononic crystals

Bloch wave:

Master equation :

Equations at macroscopic level

Effective parameters Local response: Nonlocal response: Homogenization

Si/Al 1D phononic crystals Comparison with numerical results: José A. Otero Hernández 1, Reinaldo Rodríguez 2, Julián Bravo 2 1 Instituto de Cibernética, Matemática y Física. (ICIMAF), Cuba 2 Facultad de Matemática y Computación, UH, Cuba.

Si/Al 2D phononic crystals

2D sonic crystal, solid in water (Al in water)

Comparison with: D. Torrent, J. Sánchez-Dehesa, NJP (2008):

Metamaterial response Al/Rubber 1D phononic crystal Transverse modes

Acoustic branch Local Nonlocal Local

First “optical” band Nonlocal Local Nonlocal

¡Gracias!