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2012-09-20
Dirac Dispersion and Zero-Index in Two Dimensional and Three Dimensional Photonic and Phononic Systems (Invited Paper)
By
Progress In Electromagnetics Research B, Vol. 44, 163-190, 2012
Abstract
We show that by applying accidental degeneracy, we can obtain a triply-degenerate state at the zone center in the band diagram of two dimensional (2D) photonic crystal. The dispersion near the zone center comprisestwo linear bands and an additional flat band crossing at the same frequency. If this triply-degenerate state is formed by the degeneracy of monopole and dipole excitations, we show that the system can be mapped to an effective medium with permittivity and permeability equal to zero.While "Dirac cone" dispersions can only be meaningfully defined in 2D systems, the notion of a Dirac point can be extended to three dimensional (3D) classical wave systems. We show that a simple cubic photonic crystal composed of core-shell spheres exhibitsa 3D Dirac-like point at the center ofthe Brillouin zone at a finite frequency. Using effective medium theory, we can map our structure to an isotropiczero refractive index material inwhich the effective permittivity and permeability are simultaneously zero at the Dirac-like point frequency (ωD). The Dirac-like point is six-fold degenerate and is formed by the accidental degeneracy of electric dipole and magnetic dipole excitations, each with three degrees of freedom. We found that 3D Dirac-like pointsat can also be found in simple cubic acoustic wave crystals.Different from the case in the photonic system,the 3D Dirac-like points at \overrightarrow{k}= 0 in acoustic wave systemis four-fold degenerate, and is formed by the accidental degeneracy of dipole and monopole excitations. Using effective medium theory, this acoustic wave system can also be described as a materialwhich hasboth effective mass density and reciprocal of bulk modulus equal to zero at ωD. For both the photonic and phononic systems, a subset of the bands has linear dispersions near the zone center, and they give rise to equi-frequency surfaces that are spheres with radii proportional to (ω - ωD).
Citation
Che-Ting Chan, Xueqin Huang, Fengming Liu, and Zhi Hong Hang, "Dirac Dispersion and Zero-Index in Two Dimensional and Three Dimensional Photonic and Phononic Systems (Invited Paper)," Progress In Electromagnetics Research B, Vol. 44, 163-190, 2012.
doi:10.2528/PIERB12082103
References

1. Landau, L. and E. M. Lifschitz, Electrodynamics of Continuous Media, Elsevier, New York, 1984.

2. Veselago, V. G., "The electrodynamics of substances with simultaneously negative values of ε and μ," Soviet Physics Uspekhi, Vol. 10, No. 4, 509-514, 1968.
doi:10.1070/PU1968v010n04ABEH003699

3. Pendry, J. B., A. J. Holden, D. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Transactions on Microwave Theory and Techniques, Vol. 47, No. 11, 2075-2084, 1999.
doi:10.1109/22.798002

4. Shelby, R., D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science, Vol. 292, No. 5514, 77-79, 2001.
doi:10.1126/science.1058847

5. Smith, D. R., W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Physical Review Letters, Vol. 84, 4184-4187, 2000.
doi:10.1103/PhysRevLett.84.4184

6. Zhang, S., W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Physical Review Letters, Vol. 95, 137404, 2005.
doi:10.1103/PhysRevLett.95.137404

7. Shalaev, V. M., W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials," Optics Letters, Vol. 30, No. 24, 3356-3358, 2005.
doi:10.1364/OL.30.003356

8. Dolling, G., M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780nm wavelength," Optics Letters, Vol. 32, No. 1, 53-55, 2007.
doi:10.1364/OL.32.000053

9. Pendry, J. B., "Negative refraction makes a perfect lens," Physical Review Letters, Vol. 85, 3966-3969, 2000.
doi:10.1103/PhysRevLett.85.3966

10. Garcia, N. and M. Nieto-Vesperinas, "Left-handed materials do not make a perfect lens," Physical Review Letters, Vol. 88, 207403, 2002.
doi:10.1103/PhysRevLett.88.207403

11. Grbic, A. and G. V. Eleftheriades, "Overcoming the diffraction limit with a planar left-handed transmission-line lens," Physical Review Letters, Vol. 92, 117403, 2004.
doi:10.1103/PhysRevLett.92.117403

12. Parimi, P. V., W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature, Vol. 426, 404, 2003.
doi:10.1038/426404a

13. Fang, N., H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science, Vol. 308, No. 5721, 534-537, 2005.
doi:10.1126/science.1108759

14. Chew, W. C., "Some reflections on double negative materials," Progress In Electromagnetics Research,, Vol. 51, 1-26, 2005.
doi:10.2528/PIER04032602

15. Pendry, J. B., D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science, Vol. 312, No. 5781, 1780-1782, 2006.
doi:10.1126/science.1125907

16. Leonhardt , U., "Optical conformal mapping," Science, Vol. 312, No. 5781, 1777-1780, 2006.
doi:10.1126/science.1126493

17. Schurig, D., J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science, Vol. 314, No. 5801, 977-980, 2006.
doi:10.1126/science.1133628

18. Li, J. and J. B. Pendry, "Hiding under the carpet: A new strategy for cloaking," Physical Review Letters, Vol. 101, 203901, 2008.
doi:10.1103/PhysRevLett.101.203901

19. Cheng, X., H. Chen, B. I. Wu, and J. A. Kong, "Cloak for bianisotropic and moving media," Progress In Electromagnetics Research, Vol. 89, 199-212, 2009.
doi:10.2528/PIER08120803

20. Cheng, Q., W. X. Jiang, and T. J. Cui, "Investigations of the electromagnetic properties of three-dimensional arbitrarily-shaped cloaks," Progress In Electromagnetics Research, Vol. 94, 105-117, 2009.
doi:10.2528/PIER09060705

21. Cheng, X., H. Chen, X.-M. Zhang, B. Zhang, and B. I. Wu, "Cloaking a perfectly conducting sphere with rotationally uniaxial nihility media in monostatic radar system," Progress In Electromagnetics Research, Vol. 100, 285-298, 2010.
doi:10.2528/PIER09112002

22. Lai, Y., J. Ng, H. Y. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, "Illusion optics: The optical transformation of an object into another object," Physical Review Letters, Vol. 102, 253902, 2009.
doi:10.1103/PhysRevLett.102.253902

23. Duan, Z. Y., B. I. Wu, S. Xi, H. S. Chen, and M. Chen, "Research process in reversed cherenkov radiation in double-negative metamaterials," Progress In Electromagnetics Research, Vol. 90, 75-87, 2009.
doi:10.2528/PIER08121604

24. Silveirinha, M. and N. Engheta, "Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials," Physical Review Letters, Vol. 97, 157403, 2006.
doi:10.1103/PhysRevLett.97.157403

25. Silveirinha, M. and N. Engheta, "Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media," Physical Review B, Vol. 75, 075119, 2007.
doi:10.1103/PhysRevB.75.075119

26. Silveirinha, M. G. and N. Engheta, "Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials," Physical Review B, Vol. 76, 245109, 2007.
doi:10.1103/PhysRevB.76.245109

27. Alu, A. and N. Engheta, "Dielectric sensing in ε-near-zero narrow waveguide channels," Physical Review B, Vol. 78, 045102, 2008.
doi:10.1103/PhysRevB.78.045102

28. Alu, A., M. G. Silveirinha, and N. Engheta, "Transmission-line analysis of ε-near-zero-filled narrow channels," Physical Review E, Vol. 78, 016604, 2008.
doi:10.1103/PhysRevE.78.016604

29. Edwards, B., A. Alu, M. G. Silveirinha, and N. Engheta, "Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects," Journal of Applied Physics, Vol. 105, No. 4, 044905, 2009.
doi:10.1063/1.3074506

30. Liu, R., Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, "Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies," Physical Review Letters, Vol. 100, 023903, 2008.
doi:10.1103/PhysRevLett.100.023903

31. Edwards, B., A. Alu, M. E. Young, M. Silveirinha, and N. Engheta, "Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide," Physical Review Letters, Vol. 100, 033903, 2008.
doi:10.1103/PhysRevLett.100.033903

32. Halterman, K. and S. Feng, "Resonant transmission of electromagnetic fields through subwavelength zero-ε slits," Physical Review A, Vol. 78, 021805, 2008.
doi:10.1103/PhysRevA.78.021805

33. Ziolkowski, R. W., "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Physical Review E, Vol. 70, 046608, 2004.
doi:10.1103/PhysRevE.70.046608

34. Enoch, S., G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission," Physical Review Letters, Vol. 89, 213902, 2002.
doi:10.1103/PhysRevLett.89.213902

35. Alu, A., M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern," Physical Review B, Vol. 75, 155410, 2007.
doi:10.1103/PhysRevB.75.155410

36. Hao, J., W. Yan, and M. Qiu, "Super-reflection and cloaking based on zero index metamaterial," Applied Physics Letters, Vol. 96, No. 10, 101109, 2010.
doi:10.1063/1.3359428

37. Jin, Y. and S. He, "Enhancing and suppressing radiation with some permeability-near-zero structures," Optics Express,, Vol. 18, No. 16, 16587-16593, 2010.
doi:10.1364/OE.18.016587

38. Nguyen, V. C., L. Chen, and K. Halterman, "Total transmission and total re°ection by zero index metamaterials with defects Physical Review Letters,", Vol. 105, 233908, 2010.

39. Xu, , Y. and H. Chen, "Total reflection and transmission by epsilon-near-zero metamaterials with defects," Applied Physics Letters, Vol. 98, No. 11, 113501, 2011.
doi:10.1063/1.3565172

40. Wang, L. G., Z. G.Wang, J. X. Zhang, and S. Y. Zhu, "Realization of Dirac point with double cones in optics," Optics Letters, Vol. 34, No. 10, 1510-1512, 2009.
doi:10.1364/OL.34.001510

41. Novoselov, K. S., A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Gregorieva, and A. A. Firsov, "Electric field effect in atomically thin carbon films," Science, Vol. 306, No. 5696, 666-669, 2004.
doi:10.1126/science.1102896

42. Novoselov, K. S., A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, "Two-dimensional gas of massless dirac fermions in grapheme," Nature, Vol. 438, 197-200, 2005.
doi:10.1038/nature04233

43. Zhang, Y., Y. W. Tan, H. L. Stormer, and P. Kim, "Experimental observation of the quantum hall effect and Berry's phase in grapheme," Nature, Vol. 438, 201-204, 2005.
doi:10.1038/nature04235

44. Katsnelson, M. I., K. S. Novoselov, and A. K. Geim, "Chiral tunnelling and the Klein paradox in graphene," Nature Physics, Vol. 2, 620-625, 2006.
doi:10.1038/nphys384

45. Morozov, S. V., K. S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, "Strong suppression of weak localization in graphene," Physical Review Letters, Vol. 97, 016801, 2006.
doi:10.1103/PhysRevLett.97.016801

46. Neto, A. H. C., F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, "The electronic properties of grapheme," Reviews of Modern Physics, Vol. 81, 109-162, 2009.
doi:10.1103/RevModPhys.81.109

47. Geim, A. K. and A. H. MacDonald, "Graphene: Exploring carbon flatland," Physics Today, Vol. 60, No. 8, 35-41, 2007.
doi:10.1063/1.2774096

48. Geim, A. K. and K. S. Novoselov, "The rise of graphenes," Nature Materials, Vol. 6, 183-191, 2007.
doi:10.1038/nmat1849

49. Huang, X., Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, "Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials," Nature Materials, Vol. 10, 582-586, 2011.
doi:10.1038/nmat3030

50. Liu, F., X. Huang, and C. T. Chan, "Dirac cones at k = 0 in acoustic crystals and zero refractive index acoustic materials," Applied Physics Letters, Vol. 100, No. 7, 071911, 2012.
doi:10.1063/1.3686907

51. Liu, F., Y. Lai, X. Huang, and C. T. Chan, "Dirac cones at k = 0 in phononic crystals," Physical Review B, Vol. 84, 224113, 2011.
doi:10.1103/PhysRevB.84.224113

52. Plihal, M. and A. A. Maradudin, "Photonic band structure of a two-dimensional system: The triangular lattice," Physical Review B, Vol. 44, 8565, 1991.
doi:10.1103/PhysRevB.44.8565

53. Haldane, F. D. M. and S. Raghu, "Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,", Vol. 100, 013904, 2008.

54. Raghu, S. and F. D. M. Haldane, "Analogs of quantum-hall-effect edge states in photonic crystals," Physical Review A, Vol. 78, 033834, 2008.
doi:10.1103/PhysRevA.78.033834

55. Ochiai, T. and M. Onoda, "Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states," Physical Review B, Vol. 80, 155103, 2009.
doi:10.1103/PhysRevB.80.155103

56. Ochiai, T., "Topological properties of bulk and edge states in honeycomb lattice photonic crystals: The case of TE polarization," Journal of Physics: Condensed Matter, Vol. 22, No. 22, 225502, 2010.
doi:10.1088/0953-8984/22/22/225502

57. Sepkhanov, R. A., J. Nilsson, and C. W. J. Beenakker, "Proposed method for detection of the pseudospin | 1/2 Berry phase in a photonic crystal with a Dirac spectrum," Physical Review B, Vol. 78, 045122, 2008.
doi:10.1103/PhysRevB.78.045122

58. Mei, J., Y. Wu, C. T. Chan, and Z. Q. Zhang, "First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals," Physical Review B, Vol. 86, 035141, 2012.
doi:10.1103/PhysRevB.86.035141

59. Sepkhanov, R. A., Y. B. Bazaliy, and C. W. J. Beenakker, "Extremal transmission at the Dirac point of a photonic band structure," Physical Review A, Vol. 75, 063813, 2007.
doi:10.1103/PhysRevA.75.063813

60. Diem, M., T. Koschny, and C. M. Soukoulis, "Transmission in the vicinity of the Dirac point in hexagonal photonic crystals," Physica B, Vol. 405, 2990-2995, 2010.
doi:10.1016/j.physb.2010.01.020

61. Zhang, X., "Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal," Physical Review Letters, Vol. 100, 113903, 2008.
doi:10.1103/PhysRevLett.100.113903

62. Zhang, X. and Z. Liu, "Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals," Physical Review Letters, Vol. 101, 264303, 2008.
doi:10.1103/PhysRevLett.101.264303

63. Wang, L. G., Z. G. Wang, and S. Y. Zhu, "Zitterbewegung of optical pulses near the dirac point inside a negative-zero-positive index metamaterial," EPL, Vol. 86, 47008, 2009.
doi:10.1209/0295-5075/86/47008

64. Sakoda, K., "Dirac cone in two- and three-dimensional metamaterials," Optics Express, Vol. 20, No. 4, 3898-3912, 2012.
doi:10.1364/OE.20.003898

65. Sakoda, K. and H. Zhou, "Role of structural electromagnetic resonances in a steerable left-handed antenna," Optics Express, Vol. 18, No. 26, 27371-27386, 2010.
doi:10.1364/OE.18.027371

66. Sakoda, K. and H. Zhou, "Analytical study of two-dimensional degenerate metamaterial antennas," Optics Express, Vol. 19, No. 15, 13899-13921, 2011.
doi:10.1364/OE.19.013899

67. Sakoda, K., "Double Dirac cones in triangular-lattice metamaterials," Optics Express, Vol. 20, No. 9, 9925-9939, 2012.
doi:10.1364/OE.20.009925

68. Inui, T., Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics, Springer, Berlin, 1990.
doi:10.1007/978-3-642-80021-4

69. Wu, Y., J. Li, Z. Q. Zhang, and C. T. Chan, "Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit," Physical Review B, Vol. 74, 085111, 2006.
doi:10.1103/PhysRevB.74.085111

70. Sakoda, K., Optical Properties of Photonic Crystals, 2nd Ed., Springer-Verlag, Berlin, 2004.

71. Li, J. and C. T. Chan, "Double-negative acoustic metamaterial," Physical Review E, Vol. 70, 055602, 2004.
doi:10.1103/PhysRevE.70.055602

72. Wu, Y. and Z. Q. Zhang, "Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions," Physical Review B, Vol. 79, 195111, 2009.
doi:10.1103/PhysRevB.79.195111