## From scattering or impedance matrices to Bloch modes of photonic crystals

JOSA A, Vol. 19, Issue 8, pp. 1547-1554 (2002)

http://dx.doi.org/10.1364/JOSAA.19.001547

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### Abstract

The dispersion relation of Bloch waves is derived from the properties of a single grating layer. A straightforward way to impose the Bloch condition leads to the calculation of the eigenvalues of the transfer matrix through the single grating layer. Unfortunately, the transfer-matrix algorithm is known to be unstable as a result of the growing evanescent waves. This problem appears again in the calculation of the eigenvalues, making unusable the transfer matrix in numerous practical problems. We propose two different algorithms to circumvent this problem. The first one takes advantage of scattering matrices, while the second one takes advantage of impedance matrices. Numerical evidence of the efficiency of the algorithms is given. Dispersion diagrams of simple cubic and woodpile photonic crystals are obtained by using, respectively, the scattering and impedance matrices.

© 2002 Optical Society of America

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(260.2110) Physical optics : Electromagnetic optics

**History**

Original Manuscript: September 24, 2001

Revised Manuscript: January 7, 2002

Manuscript Accepted: February 7, 2002

Published: August 1, 2002

**Citation**

Boris Gralak, Stefan Enoch, and Gérard Tayeb, "From scattering or impedance matrices to Bloch modes of photonic crystals," J. Opt. Soc. Am. A **19**, 1547-1554 (2002)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-19-8-1547

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### References

- K. Ho, C. Chan, C. Soukoulis, “Existence of photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]
- J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).
- J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–229 (1994). [CrossRef]
- A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995). [CrossRef]
- A. Moroz, C. Sommers, “Photonic band gaps of three-dimensional face-centered cubic lattices,” J. Phys. Condens. Matter 11, 997–1008 (1999). [CrossRef]
- R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999). [CrossRef]
- B. Gralak, S. Enoch, G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000). [CrossRef]
- L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001). [CrossRef]
- A. Modinos, N. Stefanou, V. Yannopapas, “Applications of the layer-KKR method to photonic crystals,” Opt. Express 8, 197–202 (2001). [CrossRef] [PubMed]
- D. M. Whittaker, “Inhibited emission in photonic woodpile lattices,” Opt. Lett. 25, 779–781 (2000). [CrossRef]
- R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. Theor. Instrum. 6, 263–281 (1963).
- P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979). [CrossRef]
- M. Nevière, R. Reinisch, D. Maystre, “Surface enhanced second harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985). [CrossRef]
- D. Maystre, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.
- E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000). [CrossRef]
- E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001). [CrossRef]
- L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981). [CrossRef]
- L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981). [CrossRef]
- S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993). [CrossRef]
- L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993). [CrossRef]
- L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996). [CrossRef]
- S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998). [CrossRef]
- J. G. Fleming, S.-Y. Lin, “Three-dimensional photonic crystal with a stop band from 1.35 to 1.95 µm,” Opt. Lett. 24, 49–51 (1999). [CrossRef]
- L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997). [CrossRef]
- S.-Y. Lin, J. G. Fleming, R. Lin, M. M. Sigalas, R. Biswas, K. M. Ho, “Complete three-dimensional photonic bandgap in a simple cubic structure,” J. Opt. Soc. Am. B 18, 32–35 (2001). [CrossRef]
- H. Sözüer, J. Haus, R. Inguva, “Photonic bands: convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992). [CrossRef]
- F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994). [CrossRef]
- D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994). [CrossRef]
- R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
- L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]

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