## Plane-wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers

JOSA A, Vol. 21, Issue 9, pp. 1769-1775 (2004)

http://dx.doi.org/10.1364/JOSAA.21.001769

Enhanced HTML Acrobat PDF (819 KB)

### Abstract

We present a new algorithm for calculation of the band structure of photonic crystal slabs. This algorithm combines the plane-wave expansion method with perfectly matched layers for the termination of the computational region in the direction out of the plane. In addition, the effective-medium tensor is applied to improve convergence. A general complex eigenvalue problem is then obtained. Two criteria are presented to distinguish the guided modes from the PML modes. As such, this scheme can accurately determine the band structure both above and below the light cone. The convergence of the algorithm presented has been studied. The results obtained by using this algorithm have been compared with those obtained by the finite-difference time-domain method and found to agree very well.

© 2004 Optical Society of America

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(130.3130) Integrated optics : Integrated optics materials

(230.7400) Optical devices : Waveguides, slab

(260.2030) Physical optics : Dispersion

(310.2790) Thin films : Guided waves

**History**

Original Manuscript: January 16, 2004

Manuscript Accepted: April 28, 2004

Published: September 1, 2004

**Citation**

Shouyuan Shi, Caihua Chen, and Dennis W. Prather, "Plane-wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers," J. Opt. Soc. Am. A **21**, 1769-1775 (2004)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-9-1769

Sort: Year | Journal | Reset

### References

- H. Y. Ryu, H. G. Park, Y. H. Lee, “Two-dimensional photonic crystal semiconductor lasers: Computational design, fabrication, and characterization,” IEEE J. Sel. Top. Quantum Electron. 8, 891–908 (2002). [CrossRef]
- R. Shimada, A. L. Yablonskii, S. G. Tikhodeev, T. Ishihara, “Transmission properties of a two-dimensional photonic crystal slab with an excitonic resonance,” IEEE J. Quantum Electron. 38, 872–879 (2002). [CrossRef]
- T. Baba, A. Motegi, T. Iwai, N. Fukaya, Y. Watanabe, A. Sakai, “Light propagation characteristics of straight single-line-defect waveguides in photonic crystal slabs fabricated into a silicon-on-insulator substrate,” IEEE J. Quantum Electron. 38, 743–752 (2002). [CrossRef]
- A. Chutinan, S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000). [CrossRef]
- M. Notomi, A. Shinya, E. Kuramochi, I. Yokohama, C. Takahashi, K. Yamada, J. Takahashi, T. Kawashima, S. Kawakami, “Si-based photonic crystals and photonic-bandgap waveguides,” IEICE Trans. Electron. E85-C, 1025–1032 (2002).
- H. G. Park, J. K. Hwang, J. Huh, H. Y. Ryu, S. H. Kim, J. S. Kim, Y. H. Lee, “Characteristics of modified single-defect two-dimensional photonic crystal lasers,” IEEE J. Quantum Electron. 38, 1353–1365 (2002). [CrossRef]
- S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). [CrossRef]
- A. Scherer, O. Painter, J. Vuckovic, M. Loncar, T. Yoshie, “Photonic crystals for confining, guiding, and emitting light,” IEEE Trans. Nanotech. 1, 4–11 (2002). [CrossRef]
- M. Loncar, T. Doll, J. Vuckovic, A. Scherer, “Design and fabrication of silicon photonic crystal optical waveguides,” J. Lightwave Technol. 18, 1402–1411 (2000). [CrossRef]
- R. D. Meade, A. M. Rappe, K. M. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic bandgap materials,” Phys. Rev. B 48, 8434–8437 (1993). [CrossRef]
- K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]
- S. G. Johnson, J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis,” Opt. Express 8, 173–180 (2001). [CrossRef] [PubMed]
- S. G. Johnson, A. Mekis, J. Fan, J. D. Joannopoulos, “Molding the flow of light,” Comput. Sci. Eng. 3, 38–47 (2001). [CrossRef]
- L. F. Shen, S. L. He, L. Wu, “The application of effective-medium theory in the plane-wave expansion method for analyzing photonic crystals,” Acta Phys. Sin. 51, 1133–1138 (2002).
- C. T. Chan, Q. L. Yu, K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995). [CrossRef]
- S. S. Xiao, L. F. Shen, S. L. He, “A plane-wave expansion method based on the effective medium theory for calculating the band structure of a two-dimensional photonic crystal,” Phys. Lett. A 313, 132–138 (2003). [CrossRef]
- P. M. Bell, J. B. Pendry, L. M. Moreno, A. J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Commun. 85, 306–322 (1995). [CrossRef]
- M. M. Sigalas, R. Biswas, K. M. Ho, C. M. Soukoulis, “Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,” Phys. Rev. B 58, 6791–6794 (1998). [CrossRef]
- M. Qiu, S. L. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000). [CrossRef]
- M. Qiu, S. L. He, “FDTD algorithm for computing the off-plane band structure in a two-dimensional photonic crystal with dielectric or metallic inclusions,” Phys. Lett. A 278, 348–354 (2001). [CrossRef]
- L. Wu, S. L. He, “Revised finite-difference time-domain algorithm in a nonorthogonal coordinate system and its application to the computation of the band structure of a photonic crystal,” J. Appl. Phys. 91, 6499–6506 (2002). [CrossRef]
- W. Axmann, P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic bandgap materials,” J. Comput. Phys. 150, 467–481 (1999). [CrossRef]
- D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999). [CrossRef]
- D. C. Dobson, J. Gopalakrishnan, J. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668–679 (2000). [CrossRef]
- J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]
- H. Rogier, D. D. Zutter, “Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer,” IEEE Trans. Microwave Theory Tech. 49, 712–715 (2001). [CrossRef]
- Q. Cao, P. Lalanne, J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]
- T. Tischler, W. Heinrich, “The perfectly matched layer as lateral boundary in finite-difference transmission-line analysis,” IEEE Trans. Microwave Theory Tech. 48, 2249–2253 (2000). [CrossRef]
- H. Derudder, F. Olyslager, D. D. Zutter, S. V. D. Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas Propag. 49, 185–195 (2001). [CrossRef]
- Z. S. Sacks, D. M. Kingsland, R. Lee, J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995). [CrossRef]
- D. E. Aspnes, “Effective medium theory,” Am. J. Phys. 50, 704–709 (1982). [CrossRef]
- A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston, Mass., 2000).
- Z. Y. Li, J. Wang, B. Y. Gu, “Creation of partial band gaps in anisotropic photonic-bandgap structures,” Phys. Rev. B 58, 3721–3729 (1998). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.