International Jubilee Seminar Current Problems in Solid State
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
Photonic crystal Photonic metamaterial
Refraction index
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 kp Sp k װ kn Sn
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
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) Homogenization or mean-field theory Rapid oscillations of fields are smoothed out:
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
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
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 and Orthorhombic PCs
Cubic lattices
Cubic lattices
Metallic wires f = 0. 001 r/a = 0. 017 p = cμ 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 1000+10 i 0 0. 1 0. 2
Left-handed metamaterial y z x
Left-handed metamaterial
Magnetometallic PC
300+5 i 1000+10 i
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:
Nonlocal dielectric response Expansion in small wave vectors (ka<< 1): Magneto-dielectric response Bianisotropic response
3 D crosses of continous wires
3 D crosses of cut wires New J. Phys. (2011)
3 D crosses of cut wires
Continuous wires Cut wires
3 D 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 55 th 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:
Photonic crystal Photonic metamaterial 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 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 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
Bloch wave:
Master equation:
Equations at macroscopic level
Effective parameters Nonlocal response: Local response: Homogenization
Si/Al 1 D 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 2 D phononic crystals
2 D sonic crystal, solid in water (Al in water)
Comparison with: D. Torrent, J. Sánchez-Dehesa, NJP (2008):
Metamaterial response Al/Rubber 1 D phononic crystal Transverse modes
Acoustic branch Local Nonlocal
First “optical” band Local Nonlocal
¡Gracias!
- Slides: 78