volume | PIER Journals

Abstract

In this paper, the properties of photonic band gaps (PBGs) for three-dimensional (3D) photonic crystals (PCs) composed of isotropic positive-index materials and epsilon-negative materials with pyrochlore lattices are theoretically investigated by a modified plane wave expansion method. The eigenvalue equations of calculating the band structure for such 3D PCs in the first irreducible Brillouin zone (spheres with the isotropic positive-index materials inserted in the epsilon-negative materials background) are theoretically deduced. Numerical simulations show that the PBG and a flatbands region can be achieved. It is also found that the larger PBG can be obtained in such PCs structure than the conventional lattices, such as diamond, face-centered-cubic, body-centered-cubic and simple-cubic lattices. The influences of the relative dielectric constant of spheres, filling factor, electronic plasma frequency, dielectric constant of epsilon-negative materials and damping factor on the properties of the PBG for such 3D PCs are studied in detail, respectively, and some corresponding physical explanations are also given. The calculated results also show that the PBG can be manipulated by the parameters mentioned above except for the damping factor. Introducing the epsilon-negative materials into 3D dielectric PCs can obtain the complete and larger PBGs as such 3D PCs with pyrochlore lattices, and also provides a way to design the potential devices.

1. Yablonovitch, E., "Inhibited spontaneous emission of photons in solid-state physics and electronics," Phys. Rev. Lett., Vol. 58, 2059-2061, 1987.
doi:10.1103/PhysRevLett.58.2059

2. John, S., "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett., Vol. 58, 2486-2489, 1987.
doi:10.1103/PhysRevLett.58.2486

3. Rybin, M. V., A. B. Khanikaev, M. Inoue, K. B. Samusev, M. J. Steel, G. Yushin, and M. F. Limonov, "Fano resonance between Mie and Bragg scattering in photonic crystals," Phys. Rev. Lett., Vol. 103, 023901-1-023901-4, 2009.
doi:10.1103/PhysRevLett.103.023901

4. Knight, J. C., J. Broeng, T. A. Birks, and P. S. J. Russell, "Photonic band gap guidance in optical fibers," Science, Vol. 284, 1476-1478, 1999.

5. Painter, O., R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic bandgap defect mode laser," Science, Vol. 284, 1819-1821, 1999.
doi:10.1126/science.284.5421.1819

6. Miyai, E., M. Okano, M. Mochizuki, and S. Noda, "Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide," Appl. Phy. Lett., Vol. 81, 3729-3731, 2002.
doi:10.1063/1.1521586

7. Happ, T. D., M. Kamp, A. Forchel, J. L. Gentner, and L. Goldstein, "Two-dimensional photonic crystal couple-defect laser diode," Appl. Phy. Lett., Vol. 82, 4-6, 2003.
doi:10.1063/1.1527703

8. Zhang, H. F., M. Li, and S. B. Liu, "Defect mode properties of magnetized plasma photonic crystals," Acta Phys. Sin., Vol. 58, 1071-1076, 2009.

9. Zhang, H. F., S. B. Liu, X. K. Kong, L. Zou, C. Z. Li, and W. S. Qing, "Enhancement of omnidirectional photonic band gaps in one-dimensional dielectric plasma photonic crystals with a matching layer," Phys. Plasmas, Vol. 19, 022103, 2012.
doi:10.1063/1.3680628

10. Zhang, H. F., S. B. Liu, X. K. Kong, B. R. Bian, and Y. Dai, "Omnidirectional photonic band gap enlarged by one-dimensional ternary unmagnetized plasma photonic crystals based on a new Fibonacci quasiperiodic structure," Phys. Plasmas, Vol. 19, 22102, 2012.
doi:10.1063/1.3680633

11. Kwon, S. H., H. Y. Ryu, G. H. Kim, Y. H. Lee, and S. B. Kim, "Photonic bandedge lasers in a two-dimensional square-lattice photonic crystal slab," Appl. Phys. Lett., Vol. 83, 3872-3879, 2002.

12. Monat, C., C. Seassal, X. Letartre, P. Regreny, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d'Yerville, D. Cassagne, J. P. Albert, and E. Jala, "InP-based two-dimensional photonic crystal on silicon: In-plane Bloch mode laser," Appl. Phys. Lett., Vol. 81, 5102-5104, 2002.
doi:10.1063/1.1532554

13. Hattori, H. T., I. McKerracher, H. H. Tan, C. Jagadish, R. Michael, and D. L. Rue, "In-plane coupling of light from InP-based photonic crystal band-edge lasers into single-mode waveguides," IEEE J. Quantum Electron, Vol. 43, 279-286, 2007.
doi:10.1109/JQE.2006.890402

14. Zhang, , H. Y., Y. P. Zhang, W. H. Liu, Y. Q. Wang, J. G. Yang, "Zero-averaged refractive-index gaps extension by using photonic heterostructures containing negative-index materials," Appl. Phys. B, Vol. 96, 67-70, 2009.
doi:10.1007/s00340-009-3419-x

15. Deng, X. H., J. T. Liu, J. H. Huang, L. Zou, and N. H. Liu, "Omnidirectional bandgaps in Fibonacci quasicrystals containing single-negative materials," J. Phys.: Condens. Matter, Vol. 22, 055403, 2010.
doi:10.1088/0953-8984/22/5/055403

16. Mocella, V., S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, "Self collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial," Phys. Rev. Lett., Vol. 102, 122902, 2009.
doi:10.1103/PhysRevLett.102.133902

17. Kocaman, S., M. S. Aras, P. Hsieh, J. F. McMillan, C. G. Biris, N. C. Panoiu, M. B. Yu, D. L. Kwong, A. Stein, and C. W. Wong, "Zero phanse delay in negative-refractive-index photonic crystals supperlattices," Nature Photonics, Vol. 5, 499-505, 2011.
doi:10.1038/nphoton.2011.129

18. Veselago, V. G., "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Uspekhi, Vol. 10, 509-514, 1968.
doi:10.1070/PU1968v010n04ABEH003699

19. Chen, Y., "Broadband one-dimensional photonic crystal wave plate containing single-negative materials," Opt. Express, Vol. 13, 19920-19929, 2010.
doi:10.1364/OE.18.019920

20. Chen, Y., "Broadband wave plates: Approach from one-dimensional photonic crystals containing metamaterials," Phys. Lett. A, Vol. 375, 1156-1159, 2011.
doi:10.1016/j.physleta.2011.01.020

21. Wang, L. G., H. Chen, and S. Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials," Phys. Rev. B, Vol. 70, 245102-1-245102-6, 2004.

22. Zhang, H. F., S. B. Liu, X. K. Kong, B. R. Bian, and Y N Guo, "Dispersion properties of two-dimensioanl plasma photonic crystals with periodically external magnetic field," Solid State Commun., Vol. 152, 1221-1229, 2012.
doi:10.1016/j.ssc.2012.04.055

23. Zhang, H. F., X. K. Kong, and S. B. Liu, "Analsys of the properties of tunable prohibited band gaps for two-dimensional unmagnetized plasma photonic crystals under TM mode," Acta Phys. Sin., Vol. 60, 055209, 2011.

24. Zhang, H. F., S. B. Liu, and X. K. Kong, "Defect mode properties of two-dimensional unmagnetized plasma photonic crystals with line-defect under transverse magnetic mode," Acta Phys. Sin., Vol. 60, 025215, 2011.

25. Zhang, H. F., S. B. Liu, X. K. Kong, B. R. Bian, and X. Zhao, "Properties of omnidirectional photonic band gaps in ¯bonacci quasi-periodic one-dimensional superconductor photonic crystals," Progress In Electromagnetics Research B, Vol. 40, 415-437, 2012.
doi:10.2528/PIERB12040406

26. Zhang, H. F., S. B. Liu, X. K. Kong, B. R. Bian, and Y. Dai, "Omnidirectional photonic band gaps enlarged by Fibonacci quasi-periodic one-dimensional ternary superconductor photonic crystals," Solid State Commun., Vol. 152, 2113-2119, 2012.
doi:10.1016/j.ssc.2012.09.009

27. Zhang, H. F., S. B. Liu, X. K. Kong, B. R. Bian, and B. Ma, "Enhancement of omnidirectional photonic bandgaps in one-dimensional superconductor --- Dielectric photonic crystals with a staggered structure," J. Supercond. Nov. Magn., Vol. 26, 77-85, 2013.
doi:10.1007/s10948-012-1712-0

28. Kamp, M., T. Happ, S. Mahnkopf, G. Duan, S. Anand, and A. Forchel, "Semiconductor photonic crystals for optoelectronics," Physica E, Vol. 21, 802-808, 2004.
doi:10.1016/j.physe.2003.11.122

29. Moroz, A., "Three-dimensional complete photonic bandgap structures in the visible," Phys. Rev. Lett., Vol. 83, 5274-5277, 1999.
doi:10.1103/PhysRevLett.83.5274

30. Kockaert, P., P. Tassin, I. Veretennicoff, G. V. D. Sande, and M. Tlidi, "Beyond the zero-diffraction regime in optical cavities with a left-handed material," J. Opt. Soc. Am. B, Vol. 26, B148-B155, 2009.
doi:10.1364/JOSAB.26.00B148

31. Krumbholz, N., K. Gerlach, F. Rutz, M. Koch, R. Piesiewicz, T. Kurner, and D. Mittleman, "Omnidirectional terahertz mirrors: A key element for future terahertz communication systems," Appl. Phys. Lett., Vol. 88, 202905, 2006.
doi:10.1063/1.2205727

32. Kim, K., "Polarization-dependent waveguide coupling utilizing single-negative materials," IEEE Photonics Technol. Lett., Vol. 17, 369-371, 2005.
doi:10.1109/LPT.2004.839375

33. Hsueh, W. J., C. T. Chen, and C. H. Chen, "Omnidirectional band gap in Fibonacci photonic crystals with metamaterials using a band-edge formalism," Phys. Rev. A, Vol. 78, 013836, 2008.
doi:10.1103/PhysRevA.78.013836

34. Wang, Z., C. T. Chan, W. Zhang, N. Ming, and P. Sheng, "Three-dimensional self-assembly of metal nanoparticles: Possible photonic crystal with a complete gap below the plasma frequency," Phys. Rev. B, Vol. 64, 113108, 2001.
doi:10.1103/PhysRevB.64.113108

35. Chan, C. T., W. Y. Zhang, Z. L. Wang, X. Y. Lei, D. Zheng, W. Y. Tam, and P. Sheng, "Photonicband gaps from metallo-dielectric spheres," Physica B, Vol. 279, 150-154, 2000.
doi:10.1016/S0921-4526(99)00705-X

36. Zhang, W. Y., X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, "Robust photonic band gap from tunable scatterers," Phys. Rev. Lett., Vol. 84, 2853, 2000.
doi:10.1103/PhysRevLett.84.2853

37. Zhang, H. F., S. B. Liu, and B. X. Li, "The properties of photonic band gaps for three-dimensional tunable photonic crystals with simple-cubic lattices doped by magnetized plasma," Optics & Laser Technology, Vol. 50, 93-102, 2013.
doi:10.1016/j.optlastec.2013.02.011

38. Zhang, H. F., S. B. Liu, X. K. Kong, and B. R. Bian, "The characteristics of photonic band gaps for three-dimensional unmagnetized dielectric plasma photonic crystals with simple-cubic lattice," Optic Commun., Vol. 288, 82-90, 2013.
doi:10.1016/j.optcom.2012.09.078

39. Zhang, H. F., S. B. Liu, and X. K. Kong, "Analysis of Voigt effects in dispersive properties for tunable three-dimensional face-centered-cubic magnetized plasma photonic crystals," Journal of Electromagnetic Waves and Applications, Vol. 27, No. 10, 1276-1292, 2013.
doi:10.1080/09205071.2013.805436

40. Zhang, H. F., S. B. Liu, and X. K. Kong, "Investigating the dispersive properties of the three-dimensional photonic crystals with face-centered-cubic lattices containing epsilon-negative materials," Applied Physics B, Vol. 112, 553-563, 2013.
doi:10.1007/s00340-013-5438-x

41. Zhang, H. F., S. B. Liu, and X. K. Kong, "Properties of anisotropic photonic band gaps in three-dimensional plasma photonic crystals containing the uniaxial material with di®erent lattices," Progress In Electromagnetics Research, Vol. 141, 267-289, 2013.
doi:10.2528/PIER13051703

42. Zhang, H. F., S. B. Liu, and X. K. Kong, "Investigation of Faraday effects in photonic band gap for tunable three-dimensional magnetized plasma photonic crystals containing the anisotropic material in diamond arrangement," Journal of Electromagnetic Waves and Applications, Vol. 27, No. 14, 1776-1791, 2013.
doi:10.1080/09205071.2013.823361

43. Zhang, H. F., S. B. Liu, and X. K. Kong, "Dispersion properties of three-dimensional plasma photonic crystals in diamond lattice arrangement," J. Lightwave Technol., Vol. 17, 1694-1702, 2013.
doi:10.1109/JLT.2013.2256879

44. Li, Z. Y., J. Wang, and B. Y. Gu, "Creation of partial band gaps in anisotropic photonic-band-gap strucutres," Phys. Rev. B, Vol. 58, 3721-3729, 1998.
doi:10.1103/PhysRevB.58.3721

45. Malkova, N., S. Kim, T. Dilazaro, and V. Gopalan, "Symmetrical analysis of complex two-dimensional hexagonal photonic crystals," Phys. Rev. B, Vol. 67, 125203-1-125203-6, 2003.

46. Garcia-Adeva, A. J., "Band structure of photonic crystals with the symmetry of a pyrochlore lattice," Phys. Rev. B, Vol. 73, 073107-1-073107-6, 2006.

47. Garcia-Adeva, A. J., "Band gap atlas for photonic crystals having the symmetry of the kagome and pyrochlore lattices," New J. Phys., Vol. 8, 86-1-86-16, 2006.

48. Garcia-Adeva, A. J., R. Balda, and J. Fernandez, "The density of electromagnetic modes in photonic crystals based on the pyrochlore and kagome lattices," Optic. Material, Vol. 27, 1733-1742, 2005.
doi:10.1016/j.optmat.2004.11.050

49. Lou, M., Q. H. Liu, and Z. Li, "Spectral element method for band structures of three-dimensional anisotropic photonic crystals," Phys. Rev. E, Vol. 80, 056702, 2012.

50. Marrone, M., V. F. Rodriguez-Esquerre, and H. E. Hernandez-Figueroa, "Novel numerical method for the analysis of 2-D photonic crystals: The cell method," Opt. Exp., Vol. 10, 1299-1304, 2002.
doi:10.1364/OE.10.001299

51. Jun, S., Y. S. Cho, and S. Im, "Moving least-square method for the band-structure calculation of 2D photonic crystals," Opt. Exp., Vol. 11, 541-551, 2003.
doi:10.1364/OE.11.000541

52. Zhang, H. F., S. B. Liu, X. K. Kong, L. Zhou, C. Z. Li, and B. R. Bian, "Comment on `Photonic bands in two-dimensional microplasma array. I. Theoretical derivation of band structures of electromagnetic wave'," J. Appl. Phys., Vol. 101, 073304, 2007; J. Appl. Phys., Vol. 110, 026104, 2011.

53. Kuzmiak, V. and A. A. Maradudin, "Photonic band structure of one-and two-dimensional periodic systems withmetallic components in the presence of dissipation," Phy. Rev. B, Vol. 55, 7427-7444, 1997.
doi:10.1103/PhysRevB.55.7427

54. Cassagne, D., C. Jouanin, and D. Bertho, "Hexagonal photonic-band-gap structures," Phys. Rev. B, Vol. 53, 7134-7142, 1996.
doi:10.1103/PhysRevB.53.7134

55. Chern, R., C. C. Chang, and C. C. Chang, "Analysis of surface plasmon modes and band structures for plasmonic crystals in one and two dimensions," Phys. Rev. E, Vol. 73, 036605-1-036605-16, 2006.

56. Zhang, H. F., S. B. Liu, and X. K. Kong, "Study of the dispersive properties of three-dimensional photonic crystals with diamond lattices containing metamaterials," Laser Phys., Vol. 23, 105815-1-105815-9, 2013.

57. Zhang, H. F., S. B. Liu, and X. K. Kong, "The properties of photonic band gaps for three-dimensional plasma photonic crystals in a diamond structure," Phys. Plasmas, Vol. 20, 042110-1-042110-1, 2013.

Prof. Hai Feng Zhang

Dr. Shaobin Liu

Dr. Hui-Chao Zhao