volume | PIER Journals

Abstract

We show that the optical modes of a periodic nanostructure with frequency dependent dielectric constant (i.e., a dispersive optical nanostructure), in general can be written as an ordinary eigenvalue problem of a "dielectric function operator", for each distinct symmetry representation of the periodic nanostructure. For a frequency dependence in the form of polynomial rational function, the problem translates to a polynomial eigenvalue equation in the frequency of the mode. The resulting problem can be solved using the basis functions of a dielectric backbone structure, which has a frequency independent dielectric constant. Rapid convergence is achieved when the basis functions are selected to be the modes of a dielectric backbone structure that minimizes the frequency perturbation of the dielectric function of the optical nanostructure. In particular, using a two dimensional photonic crystal constructed with a polar crystal as an example, we demonstrate that, remarkable simple cubic equations are sufficient to obtain accurate descriptions of eigenfrequencies.

1. Cusack, M. A., P. R. Briddon, and M. Jaros, "Absorption spectra and optical transitions in InAs/GaAs self-assembled quantum dots," Phys. Rev. B, 4047-4050, 1997.
doi:10.1103/PhysRevB.56.4047

2. Politano, A., R. G. Agostino, E. Colavita, V. Formoso, and G. Chiarello, "Electronic properties of self-assembled quantum dots of sodium on Cu(1 1 1) and their interaction with water," Surf. Sci., Vol. 601, 2656-2659, 2007.
doi:10.1016/j.susc.2006.11.079

3. Politano, A., A. R. Marino, V. Formoso, D. Farias, R. Miranda, and G. Chiarello, "Evidence for acoustic-like plasmons on epitaxial graphene on Pt(111)," Phys. Rev. B, Vol. 84, 033401, 2011.
doi:10.1103/PhysRevB.84.033401

4. Borca, B., S. Barja, M. Garnica, M. Minniti, A. Politano, J. M. Rodriguez-Garcia, J. J. Hinarejos, D. Farias, A. L. Vazquez de Parga, and , "Electronic and geometric corrugation of periodically rippled, self-nanostructured graphene epitaxially grown on Ru(0001)," New J. Phys., Vol. 12, 093018, 2010.
doi:10.1088/1367-2630/12/9/093018

5. Ariga, K., A. Vinu, Y. Yamauchi, Q. Ji, and J. P. Hill, "Nanoarchitectonics for mesoporous materials," Bull. Chem. Soc. Jpn., Vol. 85, 1-32, 2012.
doi:10.1246/bcsj.20110162

6. Xuan, W., C. Zhu, Y. Liu, and Y. Cui, "Mesoporous metal-organic framework materials," Chem. Soc. Rev., Vol. 41, 1677-1695, 2012.
doi:10.1039/c1cs15196g

7. 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

8. Yablonovitch, E., "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett., Vol. 58, 2059-2062, 1987.
doi:10.1103/PhysRevLett.58.2059

9. Joannopoulus, J. D., R. D. Meade, and J. N. Winn, Photonic Crystals: Moding The Flow of Light, Princeton University Press, Princeton, NJ, 1995.

10. Sakoda, K., Optical Properties of Photonic Crystals, Spinger, New York, 2001.
doi:10.1007/978-3-662-14324-7

11. Smith, D. R., J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science, Vol. 305, 788-792, 2004.
doi:10.1126/science.1096796

12. Liu, Y. and X. Zhang, "Metamaterials: A new frontier of science and technology," Chem. Soc. Rev., Vol. 40, 2494-2507, 2011.
doi:10.1039/c0cs00184h

13. Kosaka, H., T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B , Vol. 58, R10096-R10099, 1998.
doi:10.1103/PhysRevB.58.R10096

14. Meier, M., A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett., Vol. 74, 7-9, 1999.
doi:10.1063/1.123116

15. Politano, A., "Influence of structural and electronic properties on the collective excitations of Ag/Cu(111)," Plasmonics, Vol. 7, 131-136, 2012.
doi:10.1007/s11468-011-9285-5

16. Deubel, M., G. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater., Vol. 3, 444-447, 2004.
doi:10.1038/nmat1155

17. Chan, C. T., Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B, Vol. 51, 16635, 1995.
doi:10.1103/PhysRevB.51.16635

18. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method, 3rd Ed., Artech House, Inc., Norwood, 2005.

19. Pendry, J. B., "Photonic band structures," J. Mod. Opt., , Vol. 41, 209-229, 1994.
doi:10.1080/09500349414550281

20. Stefanou, N., V. Yannopapas, and A. Modinos, "Heterostructures of photonic crystals: Frequency bands and transmission coeffcients," Comp. Phys. Comm., Vol. 113, 49-77, 1998.
doi:10.1016/S0010-4655(98)00060-5

21. Li, Z. Y. and L. L. Lin, "Photonic band structures solved by a plane-wave-based transfer-matrix method," Phys. Rev. E, Vol. 67, 046607, 2003.
doi:10.1103/PhysRevE.67.046607

22. Deinega, A., S. Belousov, and I. Valuev, "Hybrid transfer-matrix FDTD method for layered periodic structures," Opt. Lett., Vol. 34, 860-862, 2009.
doi:10.1364/OL.34.000860

23. Hsue, Y.-C. and T.-J. Yang, "Applying a modified plane-wave expansion method to the calculations of transmissivity and reflectivity of a semi-infinite photonic crystal," Phys. Rev. E, Vol. 70, 016706, 2004.
doi:10.1103/PhysRevE.70.016706

24. Shi, S., C. Chen, and D. W. Prather, "Revised plane wave method for dispersive material and its application to band structure calculations of photonic crystal slabs," Appl. Phys. Lett., Vol. 86, 043104, 2005.
doi:10.1063/1.1855425

25. Gu, B.-Y., L.-M. Zhao, and Y.-C. Hsue, "Applications of the expanded basis method to study the properties of photonic crystals with frequency-dependent dielectric functions and dielectric losses," Physics Letters A, Vol. 355, 134-141, 2006.
doi:10.1016/j.physleta.2006.02.011

26. Yuan, J. and Y. Y. Lu, "Photonic bandgap calculations with Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A, Vol. 23, 3217-3222, 2006.
doi:10.1364/JOSAA.23.003217

27. Yuan, J., Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comp. Phys., Vol. 227, 4617-4629, 2008.
doi:10.1016/j.jcp.2008.01.014

28. Ho, K.-M., C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett., Vol. 65, 3152-3155, 1990.
doi:10.1103/PhysRevLett.65.3152

29. Johnson, S. G. and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Exp., Vol. 8, 173-190, 2001.
doi:10.1364/OE.8.000173

30. Notomi, M., "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B, Vol. 62, 10696-10705, 2000.
doi:10.1103/PhysRevB.62.10696

31. Foteinopoulou, S. and C. M. Soukoulis, "Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects," Phys. Rev. B, Vol. 72, 165112, 2005.
doi:10.1103/PhysRevB.72.165112

32. Jiang, W., R. T. Chen, and X. Lu, "Theory of light refraction at the surface of a photonic crystal," Phys. Rev. B, Vol. 71, 245115, 2005.
doi:10.1103/PhysRevB.71.245115

33. Santamara, , F. G., J. F. G. Lopez, P. V. Braun, and C. Lopez, "Optical diffraction and high-energy features in three-dimensional photonic crystals," Phys. Rev. B, Vol. 71, 195112, 2005.
doi:10.1103/PhysRevB.71.195112

34. Serebryannikov, A. E., T. Magath, and K. Schuenemann, "Bragg transmittance of s-polarized waves through finite-thickness photonic crystals with a periodically corrugated interface," Phys. Rev. E, Vol. 74, 066607, 2006.
doi:10.1103/PhysRevE.74.066607

35. Li, Z. Y., L. L. Lin, and Z. Q. Zhang, "Spontaneous emission from photonic crystals: Full vectorial calculations," Phys. Rev. Lett., Vol. 84, 4341-4344, 2000.
doi:10.1103/PhysRevLett.84.4341

36. Zhou, Y. S., X. H. Wang, B. Y. Gu, and F. H. Wang, "Photonic band gap effects on spontaneous emission lifetimes of an assembly of atoms in two-dimensional photonic crystals," Phys. Rev. E, Vol. 72, 017601, 2005.
doi:10.1103/PhysRevE.72.017601

37. Kuzmiak, V., A. A. Maradudin, and F. Pincemin, "Photonic band structures of two-dimensional systems containing metallic components," Phys. Rev. B, Vol. 50, 16835-16844, 1994.
doi:10.1103/PhysRevB.50.16835

38. Kuzmiak, V. and A. A. Maradudin, "Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation," Phys. Rev. B, Vol. 55, 7427-7444, 1997.
doi:10.1103/PhysRevB.55.7427

39. Kuzmiak, V., A. A. Maradudin, and A. R. McGurn, "Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal," Phys. Rev. B, Vol. 55, 4298-4311, 1997.
doi:10.1103/PhysRevB.55.4298

40. Zhang, W., A. Hu, X. Lei, N. Xu, and N. Ming, "Photonic band structures of a two-dimensional ionic dielectric medium," Phys. Rev. B, Vol. 54, 10280-10283, 1996.
doi:10.1103/PhysRevB.54.10280

41. Lee, W. M., P. M. Hui, and D. Stroud, "Propagating photonic modes below the gap in a superconducting composite," Phys. Rev. B, Vol. 51, 8634-8637, 1995.
doi:10.1103/PhysRevB.51.8634

42. Raman, A. and S. Fan, "Photonic band structure of dispersive metamaterials formulated as a hermitian eigenvalue problem," Phys. Rev. Lett., Vol. 104, 087401, 2010.
doi:10.1103/PhysRevLett.104.087401

43. Toader, O. and S. John, "Photonic band gap enhancement in frequency-dependent dielectrics," Phys. Rev. E, Vol. 70, 046605, 2004.
doi:10.1103/PhysRevE.70.046605

44. Inui, T., Y. Tanabe, and Y. Onodera, Group Theory and Its Application in Physics,, Springer, Berlin, 1996.

45. Alagappan, G., X. W. Sun, and H. D. Sun, "Symmetries of the eigenstates in an anisotropic photonic crystal," Phys. Rev. B , Vol. 77, 195117, 2008.
doi:10.1103/PhysRevB.77.195117

46. Lopez-Tejeira, F., T. Ochiai, K. Sakoda, and J. Sanchez-Dehesa, "Symmetry characterization of eigenstates in opal-based photonic crystals ," Phys. Rev. B, Vol. 65, 195110, 2002.
doi:10.1103/PhysRevB.65.195110

47. Sakoda, K., N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, "Photonic bands of metallic systems. I. Principle of calculation and accuracy ," Phys. Rev. B, Vol. 64, 045116, 2001.
doi:10.1103/PhysRevB.64.045116

48. Gohberg, I., P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, London, 1982.

49. Yariv, A., "Introduction to Theory and Applications Mechanics," John Wiley & Sons Inc., 1982.

50. Kittel, C., Introduction to Solid State Physics, 7th Ed., Wiley, 1995.

51. Halevi, P. and F. Ramos-Mendieta, "Tunable photonic crystals with semiconducting constituents," Phys. Rev. Lett., Vol. 85, 1875, 2000.
doi:10.1103/PhysRevLett.85.1875

52. Weber, M. J., Handbook of Optical Materials, CRC Press, 2002.
doi:10.1201/9781420050196

53. Deinega, A. and S. John, "Effective optical response of silicon to sunlight in the finite-difference time-domain method," Opt. Lett., Vol. 37, 112-114, 2012..
doi:10.1364/OL.37.000112

54. Ribbing, C. G., H. HogstrÄom, and A. Rung, "Studies of polaritonic gaps in photonic crystals," Appl. Opt., Vol. 45, 1575-1582, 2006.
doi:10.1364/AO.45.001575

55. Foteinopoulou, S., M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials," Phys. Rev. B, Vol. 84, 035128, 2011.
doi:10.1103/PhysRevB.84.035128

Dr. Gandhi Alagappan

Dr. Alexei Deinega