volume | PIER Journals

Abstract

Multipole methods have evolved to be an important class of theoretical and computational techniques in the study of photonic crystals and related problems. In this chapter, we present a systematic and unified development of the theory, and apply it to a range of scattering problems including finite sets of cylinders, two-dimensional stacks of grating and the calculation of band diagrams from the scattering matrices of grating layers. We also demonstrate its utility in studies of finite systems that involve the computation of the local density of states.

1. Pendry, J. B. and A. MacKinnon, Phys. Rev. Lett., Vol. 69, 2772, 1992.
doi:10.1103/PhysRevLett.69.2772 Google Scholar

2. Sigalas, M. M. and et. al., Phys. Rev. B, Vol. 52, 11744, 1995.
doi:10.1103/PhysRevB.52.11744 Google Scholar

3. Ho, K. M., C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett., Vol. 65, 3152, 1990.
doi:10.1103/PhysRevLett.65.3152 Google Scholar

4. Rayleigh, J. W. S., Philos. Mag., Vol. 34, 481, 1892. Google Scholar

5. von Ignatowsky, W., Ann. Physik, Vol. 44, 1914. Google Scholar

6. Twersky, V., Arch. Rational Mech. Anal., Vol. 8, 323, 1961. Google Scholar

7. McPhedran, R. C., et al. Aust. J. Phys., Vol. 52, 791, 1999. Google Scholar

8. Botten, L. C., et al. "Electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations Part 1: Method," J. Opt. Soc. Am. A, Vol. 17, 2165, 2000. Google Scholar

9. Botten, L. C., et al. "Electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations Part 2: Properties and implementation," J. Opt. Soc. Am. A, Vol. 17, 2177, 2000. Google Scholar

10. Modinos, A., V. Karathanos, and N. Stefanou, "Optical properties of layers and crystals of spherical particles," Appl. Surface Science, Vol. 13, 65, 1993. Google Scholar

11. Stefanou, N., V. Yannopapas, and A. Modinos, , Comput. Phys. Comm., Vol. 113, 49, 1998.
doi:10.1016/S0010-4655(98)00060-5 Google Scholar

12. Kohn, W. and N. Rostoker, Phys. Rev., Vol. 94, 1111, 1954.
doi:10.1103/PhysRev.94.1111 Google Scholar

13. Nicorovici, N. A., R. C. McPhedran, and L. C. Botten, Phys. Rev. Lett., Vol. 75, 1507, 1995.
doi:10.1103/PhysRevLett.75.1507 Google Scholar

14. Nicorovici, N. A., R. C. McPhedran, and L. C. Botten, Phys. Rev. E, Vol. 52, 1135, 1995.
doi:10.1103/PhysRevE.52.1135 Google Scholar

15. McPhedran, R. C., D. H. Dawes, L. C. Botten, and N. A. Nicorovici, J. Electromagn. Waves Applications, Vol. 10, 1083, 1996. Google Scholar

16. Botten, L. C., R. C. McPhedran, N. A. Nicorovici, and A. B. Movchan, J. Electromagn. Waves Applications, Vol. 12, 847, 1998. Google Scholar

17. Poulton, C. G., et al. "Noncommuting limits in electromagnetic scattering: asymptotic analysis for an array of highly conducting inclusions," SIAM J. Appl. Math., Vol. 61, 1706, 2001.
doi:10.1137/S0036139999352262 Google Scholar

18. Lo, K. M., et al. IEEE J. Lightwave Technol., Vol. 12, 396, 1994.
doi:10.1109/50.285321 Google Scholar

19. Felbacq, D. and et. al., J. Opt. Soc. Am. A, Vol. 11, 2526, 1994. Google Scholar

20. Chin, S. K., N. A. Nicorovici, and R. C. McPhedran, Phys. Rev. E, Vol. 49, 4590, 1994.
doi:10.1103/PhysRevE.49.4590 Google Scholar

21. Ewald, P. P., Ann. Phys., Vol. 64, 253, 1921. Google Scholar

22. McPhedran, R. C., N. A. Nicorovici, L. C. Botten, and K. A. Grubits, J. Math. Phys., Vol. 41, 7808, 2000.
doi:10.1063/1.1310361 Google Scholar

23. Asatryan, A. A., P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. Martijn de Sterke, "Effects of disorder on wave propagation in two-dimensional photonic crystals,'' Phys. Rev. E, Vol. 60, 6118, 1999; Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals," Phys. Rev. E, Vol. 62, 5711, 2000.
doi:10.1103/PhysRevE.62.5711 Google Scholar

24. Cao, H., et al. Phys. Rev. Lett., Vol. 82, 2278, 2000.
doi:10.1103/PhysRevLett.82.2278 Google Scholar

25. McPhedran, R. C., N. A. Nicorovici, L. C. Botten, and K.- D. Bao, "Green's function, lattice sum and Rayleigh's identity for a dynamic scattering problem," IMA Volumes in Mathematics and its Applications, Vol. 96, 155-186, 1997. Google Scholar

26. Born, M. and E. Wolf, Principles of Optics, University Press, 1998.

27. Wijngaard, W., J. Opt. Soc. Am., Vol. 63, 944, 1973. Google Scholar

28. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1972.

29. Oberhettinger, F., Fourier Expansions, 33, Academic, 1973.

30. SÃ¶zuer, H. S. and J. P. Dowling, J. Mod. Opt., Vol. 41, 231-239, 1994. Google Scholar

31. Line, S. H. and J. G. Fleming, IEEE J. Lightwave Technol., Vol. 17, 1944, 1999.
doi:10.1109/50.802977 Google Scholar

32. Botten, L. C., et al. Optica Acta, Vol. 28, 413, 1981. Google Scholar

33. McRae, E. G., Surface Science, Vol. 11, 479, 1968.
doi:10.1016/0039-6028(68)90058-7 Google Scholar

34. Gralak, B., S. Enoch, and G. Tayeb, J. Opt. Soc. Am. A, Vol. 17, 1012, 2000. Google Scholar

35. Parker, A. R., et al. Nature, Vol. 409, 36, 2000.
doi:10.1038/35051168 Google Scholar

36. Guida, G., Opt. Comm., Vol. 156, 294, 1998.
doi:10.1016/S0030-4018(98)00462-3 Google Scholar

37. Sigalas, M. M. and et. al., Phys. Rev. B, Vol. 53, 8340, 1996.
doi:10.1103/PhysRevB.53.8340 Google Scholar

38. Sprik, R., B. A. van Tiggelen, and A. Lagendijk, Europhys. Lett., Vol. 35, 265, 1996.
doi:10.1209/epl/i1996-00564-y Google Scholar

39. John, S. and J. Wang, Phys. Rev. B, Vol. 43, 12772, 1991.
doi:10.1103/PhysRevB.43.12772 Google Scholar

40. Busch, K. and S. John, Phys. Rev. E, Vol. 58, 3896, 1998.
doi:10.1103/PhysRevE.58.3896 Google Scholar

41. Moroz, A., Europhys. Lett., Vol. 46, 419, 1999.
doi:10.1209/epl/i1999-00278-2 Google Scholar

42. Asatryan, A. A., et al. "Two-dimensional Green function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length," Phys. Rev. E, Vol. 63, 046612, 2001.
doi:10.1103/PhysRevE.63.046612 Google Scholar

43. Balian, R. and C. Bloch, Ann. Phys., Vol. 64, 271, 1971.
doi:10.1016/0003-4916(71)90286-7 Google Scholar

44. Martin, O. J. F. and N. B. Piller, Phys. Rev. E, Vol. 58, 3909, 1998.
doi:10.1103/PhysRevE.58.3909 Google Scholar

45., http://www.netlib.org/liblist.html.

46. Joannopoulos, J. D., R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University, 1995.

47. Zhang, W. Y. and et. al., Phys. Rev. Lett., Vol. 84, 2853, 2000.
doi:10.1103/PhysRevLett.84.2853 Google Scholar

48. Knight, J. C., T. A. Birks, P. St J. Russell, and D. M. Atkin, Opt. Lett., Vol. 21, 1547, 1996. Google Scholar

49. Botten, L. C., N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E, Vol. 64, 046603, 2001.
doi:10.1103/PhysRevE.64.046603 Google Scholar

50. Asatryan, A. A., K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, "Two-dimensional Green's function and local density of states in photonic crystals, consisting of a finite number of cylinders of infinite length," Phys. Rev. E, Vol. 63, 046612, 2001.
doi:10.1103/PhysRevE.63.046612 Google Scholar