## 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

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### 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

## 1. INTRODUCTION

2. 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]

3. H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. **41**, 231–239 (1994). [CrossRef]

4. 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]

## 2. PROBLEM FORMULATION

*L*. In a Cartesian coordinate system

*x*axis or parallel to the

*y*axis, depending on the layer, and they are bounded by the two planes at

*D*is positive. In Fig. 1 , we show a woodpile structure as a special crossed array of cylinders where the axes of the cylinders in two layers separated by one intermediate layer are offset by half a period. Outside the cylinders and in the half-spaces

*z*components and derive the following system:where

**u**and

**v**are the

*x*and

*y*components, respectively, i.e.,and

*x*and

*y*derivatives:

*x*and

*y*directions and the electromagnetic field is quasi-periodic, we can formulate a boundary value problem on the square cylinderFor that purpose, we rewrite the quasi-periodic condition as Since Eq. (2) is a first-order system with respect to

*z*, the boundary conditions at

*z*derivatives of

**u**and

**v**. To write down these boundary conditions, we define two linear operators acting on quasi-periodic functions by for

*j*and

*k*, where

**u**and

**v**are continuous across the planes at

## 3. OPERATOR MARCHING METHOD

18. 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]

27. 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]

30. 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]

## 4. MATRIX APPROXIMATIONS FOR OPERATORS

*P*,

*Q*,

*X,*and

*Y*by matrices. These operators act on column vectors, of which the components are quasi-periodic functions of

*x*and

*y*. The quasi-periodic functions are associated with

*x*and

*y*components of the incident wave vector) and satisfy a condition like Eq. (7). Since the quasi-periodic functions can be expanded in Fourier series, these operators can be represented by matrices in Fourier space.

*T*be a linear operator such that if

**g**is a vector of two quasi-periodic functions, then so is

**h**=

*T*

**g**. In physical space, if one period of

*x*or

*y*is discretized by

*N*points, then

**g**and

**h**are approximated by vectors of length

*T*is approximated by a

*T*on different Fourier modes. For a given integer pair

*T*acts on vectors, we need to consider

**g**and

**h**be also expanded in Fourier series as where

*T*

**g**=

**h**implies

*j*,

*k*,

*l*, and

*m*to

*N*terms (i.e., from

*N*is even, or from

*N*is odd), then all retained 2 × 2 matrices

**g**and

**h**. We introduce a column vector

**g**with the same

*k*together, that is,Notice that

**h**. Based on this ordering, we can assemble the 2 × 2 matrices

*T*

**g**=

**h**or Eq. (28) is approximated by

## 5. THE FIRST AND LAST STEPS

*Q*and

*Y*at

*y*components of the electromagnetic field are given byfor

*z*derivative of

**v**isfor

*T*and define an operator

**v**satisfiesThe above can be applied to

*Q*,

*Y*} or {

*P*,

*X*} at

*y*components of the electromagnetic field. For

## 6. TRANSITION THROUGH AN INTERFACE

*P*or

*Q*from

*Q*. From the second equation of the first order system [Eq. (2)], we obtainwhere

**u**and

**v**at evaluated

**u**and insert it into the equation at

*μ*by

## 7. MARCHING THROUGH A LAYER

*y*axis. For this layer, we calculate the operators

*Q*and

*Y*at

*y*invariant, we can expand the field in Fourier series for the

*y*variable asfor

*y*dependence

*y*-invariant layer and a fixed integer

*k*, the DtN map method developed in our previous work [30

30. 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]

*x*direction, i.e.,

*x*derivative on the vertical edges and find the reduced DtN map

*x*by

*N*points as

*z*derivatives at

30. 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]

**26**, 1442–1449 (2009). [CrossRef]

*P*,

*X*} and {

*Q*,

*Y*}. In Fourier space,

*x*) evaluated at

*z*derivatives of these coefficients evaluated at

*N*Fourier modes are retained as before,

**v**at

*Q*and

*Y*given in Eqs. (25, 26) imply thatTherefore, Eq. (49) can be replaced bySolving

27. 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]

**26**, 1442–1449 (2009). [CrossRef]

*x*axis, we need to use the operators

*P*,

*X*and the vector

**u**. In that case, the second ordering of the Fourier coefficients given in Eq. (30) is preferred; therefore, the marching step is carried out with

## 8. SWITCHING THE OPERATORS

*Q*,

*Y*} and {

*P*,

*X*}. From the definitions of

*P*and

*Q*and the governing Eq. (2), we haveSimilar to the derivation of Eq. (51), we first solve

**v**in terms of

**u**asand then obtainIn Fourier space and following the first ordering [Eq. (29)], the above becomesThe matrices

*N*is the number of retained terms for Fourier series in

*x*or

*y*. The switching formula [Eq.(55)] is only used at the interfaces between the layers where the medium is homogeneous, i.e., ɛ and μ are constants. In that case,

*x*invariant layer uses

*x*invariant layer, it is necessary to transform

## 9. NUMERICAL EXAMPLES

*M*layers of crossed arrays of circular dielectric cylinders surrounded by air. The dielectric constant and the radius of the cylinders are ɛ = 5 and

*L*is the period in both

*x*and

*y*directions. The distance between two nearby layers is also

*L*, that is,

*x*and

*y*axes, respectively. We consider two cases. The first case is a regular crossed array of cylinders where all even (or odd) layers are identical. Therefore, the structure repeats itself every two layers. The second case is a woodpile structure, where the axes of cylinders in two layers separated by one intermediate layer are offset by

*N*at fixed frequencies. Since exact solutions are not available, we use the results obtained with a large

*N*as a reference solution for computing approximate relative errors. For

*N*is increased.

## 10. CONCLUSIONS

*N*can be quite small. Typically, three significant digits can be obtained using

*z*direction. For simplicity, the method is presented for the special case where the periods in the

*x*and

*y*directions are the same, but it can be easily extended to the case where the periods in these two directions are different. Compared with the multipole method developed in [16

16. 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]

## ACKNOWLEDGMENTS

1. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

2. | 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. |

3. | H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. |

4. | 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 |

5. | K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. |

6. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

7. | D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. |

8. | A. Taflove and S. C. Hagness, |

9. | L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

10. | 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 |

11. | M. Nevière and E. Popov, |

12. | J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. |

13. | G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A |

14. | 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. |

15. | E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A |

16. | 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 |

17. | K. Yasumoto and H. Jia, “Electromagnetic scattering from multilayered crossed-arrays of circular cylinders,” SPIE5445, 200–205 (2004). |

18. | Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. |

19. | J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A |

20. | S. Li and Y. Y. Lu, “Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells,” J. Opt. Soc. Am. A |

21. | J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. |

22. | H. Xie and Y. Y. Lu, “Modeling two-dimensional anisotropic photonic crystals by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A |

23. | J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. |

24. | Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B |

25. | S. Li and Y. Y. Lu, “Computing photonic crystal defect modes by Dirichlet-to-Neumann maps,” Opt. Express |

26. | Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. |

27. | 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 |

28. | Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express |

29. | Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. |

30. | 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 |

**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

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### 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] [PubMed]
- 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] [PubMed]
- 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] [PubMed]
- 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] [PubMed]
- 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]

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