1. “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

2. Plan 1.Metamateriales fotónicos 2.Metamateriales fonónicos

4. Photonic crystal Photonic metamaterial

5. Refraction index

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

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

8. kpkp SpSp knkn SnSn kװkװ fuente Refracción negativa

9. Simulation of refraction Pendry and Smith, Phys.Today (2004).

10. Shelby, Smith and Schultz, Science (2001) Observation of negative refraction

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

12. E. Plum, et al (2009)

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

19. 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)

20. Theory is very general: Arbitrary dielectric, metallic, magnetic, and chiral inclusions. Arbitrary Bravais lattice. Inclusions in neighboring cells can be isolated or in contact.

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

22. 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, 103520 (2009).

23. A photonic crystal being periodic by definition:

24. Master equation

25. Macroscopic fields

26. Effective parameters Homogenization

27. Cubic lattice of small spheres Maxwell Garnett

28. Cubic and Orthorhombic PCs

30. Cubic lattices

32. Metallic wires f = 0.001 r/ a = 0.017 p = c μ 0 a σ z

33. Pendry´s formula

34. Magnetic wires

35. High-permeability metals and alloys

36. Magnetic properties of various grades of iron

37. High-permeability magnetic wires z 1000+10i 0 0.10.2

39. Left-handed metamaterial x zy

41. Magnetometallic PC

42. 300+5i1000+10i

43. 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)

44. Xu et al (2005)

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

47. f=0.5/100.5

48. 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:

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

50. 3D crosses of continous wires

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

53. Continuous wires Cut wires

54. 3D crosses of asymmetrically-cut wires

57. “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

58. Plan 1.Phononic crystals 2.Homogenization theory 3.Comparison with other approaches 4.Elastic metamaterials

59. Phononic crystals  (r), C l (r), C t (r) Wave equation:

60. Photonic crystal Photonic metamaterial Phononic crystal Phononic metamaterial  eff, C t,eff C l,eff New J. Phys. 13, 073041 (2011) J. Appl. Phys 106, 103520 (2009)

61. 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:

62. Phononic metamaterials ¿How can one obtain a negative mass?

63. Resonant sonic materials Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, P. Sheng, Science, 2000.

64. Z. Yang, J. Mei, M. Yang, N. H. Chan, P. Sheng, PRL, 2008 Membrane-Type Acoustic Metamaterial with Negative Dynamic Mass

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

66. Homogenization of phononic crystals

67. Bloch wave:

68. Master equation :

69. Equations at macroscopic level

70. Effective parameters Local response: Nonlocal response: Homogenization

71. 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.

72. Si/Al 2D phononic crystals

73. 2D sonic crystal, solid in water (Al in water)

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

75. Metamaterial response Al/Rubber 1D phononic crystal Transverse modes

76. Acoustic branch Local Nonlocal Local

77. First “optical” band Nonlocal Local Nonlocal

78. ¡Gracias!