## Dirichlet-to-Neumann map method for analyzing crossed arrays of circular cylinders

JOSA B, Vol. 26, Issue 11, pp. 1984-1993 (2009)

http://dx.doi.org/10.1364/JOSAB.26.001984

Acrobat PDF (287 KB) | Spotlight

### Abstract

An efficient and accurate computational method is developed for analyzing finite layers of crossed arrays of circular cylinders, including woodpile structures as special cases. The method relies on marching a few operators (approximated by matrices) from one side of the structure to another. The marching step makes use of the Dirichlet-to-Neumann (DtN) maps for two-dimensional unit cells in each layer where the structure is invariant in the direction of the cylinder axes. The DtN map is an operator that maps two wave field components to their normal derivatives on the boundary of the unit cell, and they can be easily constructed by vector cylindrical waves. Unlike existing numerical methods for crossed gratings, our method does not require a discretization of the structure. Compared with the multipole method that uses vector cylindrical wave expansions and scattering matrices, our method is relatively simple since it does not need sophisticated lattice sums techniques.

© 2009 Optical Society of America

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: July 10, 2009

Manuscript Accepted: August 24, 2009

Published: October 5, 2009

**Virtual Issues**

October 8, 2009 *Spotlight on Optics*

**Citation**

Yumao Wu and Ya Yan Lu, "Dirichlet-to-Neumann map method for analyzing crossed arrays of circular cylinders," J. Opt. Soc. Am. B **26**, 1984-1993 (2009)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-26-11-1984

Sort: Year | Journal | Reset

### References

- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton Univ. Press, 1995).
- K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413-416 (1994). [CrossRef]
- H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231-239 (1994). [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, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998). [CrossRef]
- K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis,” Opt. Express 8, 173-190 (2001). [CrossRef]
- D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time Domain Method, 2nd ed. (Artech House, 2000).
- L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758-2767 (1997). [CrossRef]
- E. Popov and M. Nevière, “Maxwell equations in Fourier space: a fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886-2894 (2001). [CrossRef]
- M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel Dekker, 2003).
- J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002). [CrossRef]
- G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106-1114 (2005). [CrossRef]
- G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1-34 (2010).
- E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33-42 (2002). [CrossRef]
- G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003). [CrossRef]
- K. Yasumoto and H. Jia, “Electromagnetic scattering from multilayered crossed-arrays of circular cylinders,” SPIE 5445, 200-205 (2004).
- Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448-3453 (2006). [CrossRef]
- J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217-3222 (2006). [CrossRef]
- S. Li and Y. Y. Lu, “Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells,” J. Opt. Soc. Am. A 24, 2438-2442 (2007). [CrossRef]
- J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. 9, 4617-4629 (2008). [CrossRef]
- H. Xie and Y. Y. Lu, “Modeling two-dimensional anisotropic photonic crystals by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 26, 1606-1614 (2009). [CrossRef]
- J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114-120 (2007). [CrossRef]
- Y. Huang, Y. Y. Lu and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 24, 2860-2867 (2007). [CrossRef]
- S. Li and Y. Y. Lu, “Computing photonic crystal defect modes by Dirichlet-to-Neumann maps,” Opt. Express 15, 14454-14466 (2007). [CrossRef]
- Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).
- Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466-1473 (2008). [CrossRef]
- Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383-17399 (2008). [CrossRef]
- Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921-932 (2008). [CrossRef]
- Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing periodic arrays of cylinders with oblique incident waves,” J. Opt. Soc. Am. B 26, 1442-1449 (2009). [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.