## Moving least-square method for the band-structure calculation of 2D photonic crystals

Optics Express, Vol. 11, Issue 6, pp. 541-551 (2003)

http://dx.doi.org/10.1364/OE.11.000541

Acrobat PDF (645 KB)

### Abstract

The moving least-square (MLS) basis is implemented for the real-space band-structure calculation of 2D photonic crystals. A *value-periodic* MLS shape function is thus proposed in order to represent the periodicity of crystal lattice. Through numerical examples, this MLS method is proved to be a promising scheme for predicting band gaps of photonic crystals.

© 2003 Optical Society of America

## 1. Introduction

2. Y. Xia, “Photonic crystals,” Adv. Mater. **13**, 369 (2001) and papers in this special issue. [CrossRef]

3. K. Busch, “Photonic band structure theory: assesment and perspectives,” C. R. Physique **3**, 53–66 (2002). [CrossRef]

4. D. Cassagne, “Photonic band gap materials,” Ann. Phys. Fr. **23**(4), 1–91 (1998). [CrossRef]

5. J.B. Pendry, “Calculating photonic band structure,” J. Phys.: Condens. Matter **8**, 1085–1108 (1996). [CrossRef]

4. D. Cassagne, “Photonic band gap materials,” Ann. Phys. Fr. **23**(4), 1–91 (1998). [CrossRef]

6. H.S. Sözüer, J.W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B **45**, 13962–13972 (1992). [CrossRef]

7. R.D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, and O.L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B **48**, 8434–8437 (1993). [CrossRef]

8. C.T. Chan, Q.L. Yu, and K.M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B **51**, 16635–16642 (1995). [CrossRef]

9. A.J. Ward and J.B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B **58**, 7252–7259 (1998). [CrossRef]

10. M. Qiu and S. 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]

11. K.M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B **48**, 7767–7771 (1993). [CrossRef]

12. X. Wang, X.G. Zhang, Q. Yu, and B.N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B **47**, 4161–4167 (1993). [CrossRef]

13. J.B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. **69**, 2772–2775 (1992). [CrossRef] [PubMed]

14. L. Shen, S. He, and S. Xiao, “A finite-diference eigenvalue algorithm for calculating the band structure of a photonic crystal,” Comput. Phys. Comm. **143**, 213–221 (2002). [CrossRef]

15. W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case,” J. Comput. Phys. **150**, 468–481 (1999). [CrossRef]

16. D.C. Dobson, “An efficient band structure calculations in 2D photonic crystals,” J. Comput. Phys. **149**, 363–376 (1999). [CrossRef]

17. C. Mias, J.P. Webb, and R.L. Ferrari, “Finite element modelling of electromagnetic waves in doubly and triply periodic structures,” IEE Proc.-Optoelectron. , **146**(2), 111–118 (1999). [CrossRef]

18. M. Marrone, V.F. Rodriguez-Esquerre, and H.E. Hernandez-Figueroa, “Novel numerical method for the analysis of 2D photonic crystals: the cell method,” Opt. Express **10**, 1299–1304 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1299 [CrossRef] [PubMed]

19. S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Review , **55**, 1–34 (2002). [CrossRef]

## 2. The moving least-square (MLS) approximation

*u*

_{J}is assigned to each node. The approximation

*u*

_{h}(x) of a function

*u*(x) at any point x in the domain of computation is then expressed by

*NP*is the total number of nodes. By minimizing the localized error residual functional that is expressed by the moving least-square procedure (see Appendix or [19

19. S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Review , **55**, 1–34 (2002). [CrossRef]

20. D.W. Kim and Y. Kim, “Point collocation method using the fast moving least-square reproducing kernel approximation,” Int. J. Numer. Methods Engrg. **56**, 1445–1464 (2003). [CrossRef]

*N*

_{J}(x) is defined as

_{J}-x) is the polynomial basis vector and

*W*(x

_{J}-x) the window function. They are here abbreviated as p

_{J}(x)=p(x

_{J}-x) and

*W*

_{J}(x)=W(x

_{J}-x). The vector b(x) in Eq. (2) is determined by solving the matrix equation of

^{T}and the moment matrix M(x) is defined by

*compact support*) of which the center is at the position x. The comprehensive description of MLS-based meshfree methods is found in a recent review paper [19

19. S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Review , **55**, 1–34 (2002). [CrossRef]

## 3. Galerkin formulation and matrix eigen-equations

*c*)

^{2}and ε(x) is the dielectric function. ω is the frequency of the monochromatic electromagnetic wave,

*c*the speed of light, and x the position vector in x-y plane. Applying the Floquet-Bloch theorem, the problems are reduced to solving for

*u*(x) as

**k**is the wave vector and

*u*(x) is the function fulfilling the periodicity of lattice structure of which the lattice vector is

**L**, i.e.,

*u*(x)=

*u*(x+

**L**).

*P*. We assume for now that the periodic MLS shape function enables the approximated functions to satisfy the periodic boundary conditions required. The derivation of such shape function is given in the next section. The matrix eigen-equations are solved using the eigensystem subroutine package (EISPACK) for the numerical examples in Section 5.

## 4. Value-periodic MLS shape functions

*N*

_{J}(x) at a specific point x, we have to search the associated nodes denoted by

*J*due to the localized property of the shape function. These nodes are located under the support in which the shape function does not vanish. While searching those nodes, we must take the periodicity into consideration. This can be implemented step by step, as shown in Fig. 1 for the point x near the lower left corner of the parallelogram. We first find nodes (inside the unit cell) associated with the original point x. Secondly, we translate the point x to x+a

_{1}and search interior nodes again. Next, it is repeated for x→x+a

_{2}and for x→x+a

_{1}+a

_{2}. This

*translation-and-searching*procedure is carried out for all translated points at which the support of shape function covers the unit cell. The resulting supports are illustrated in gray color in Fig. 1.

**L**=

*n*

_{1}a

_{1}+

*n*

_{2}a

_{2}, the MLS approximation can be written in general form as

*N*

_{J}(x+

**L**) is defined by

**L**. The vector b(x) is obtained by solving

**L**is performed over the current unit cell itself and the neighboring cells only, because, for any farther

**L**, the corresponding support does not overlap the unit cell of computation. In conclusion, the 2D

*value-periodic*MLS shape function for a parallelogram unit cell whose primitive vectors are a

_{1}and a

_{2}, can be written as

*n*

_{i}’s results in 9 cells in 2D (or 27 cells for 3D case) at most. That is,

*n*

_{i}=-1, 0, +1 in Eq. (15) is enough for any x in the unit cell. It should be carefully noted that the opposing boundaries are physically identical, i.e., Γ

_{1}=Γ

_{3}and Γ

_{2}=Γ

_{4}as illustrated in Fig. 1. Therefore, for the nodal summation over

*J*=1, …,

*NP*, we must involve one boundary only among each identical pair in order to avoid over-summing. In other word, the nodes on Γ

_{3}and Γ

_{4}have to make no contribution to the summation with respect to

*J*, once the corresponding nodes along Γ

_{1}and Γ

_{2}are summed up. An example of 2D periodic MLS shape function is plotted in Fig. 2 for a parallelogram unit cell.

## 5. Numerical examples

### 5.1 Electromagnetic Kronig-Penney problem

17. C. Mias, J.P. Webb, and R.L. Ferrari, “Finite element modelling of electromagnetic waves in doubly and triply periodic structures,” IEE Proc.-Optoelectron. , **146**(2), 111–118 (1999). [CrossRef]

_{1}=1.0 and ε

_{2}=9.0 are used for the simulation. Frequency band structures of both TM and TE modes are given in Fig. 4. Lines are for the analytic solution and open circles for the MLS results using 11×11 nodes for TM modes and 41×41 nodes for TE modes. Convergence rates of the lowest five eigenvalues at point of

**k**=(π/

*a*)[0.5, 0.5] are given in Fig. 5 in which

*E*

_{n}’s denote the normalized frequency eigenvalues (ω

_{n}

*a*/2π

*c*) and

*E*

_{e}is the analytic solution. The average slope of TM and TE modes are 3.12 and 2.77 respectively. The results for TM modes show better performance over the TE modes case, not only in convergence but also in accuracy. This is due to the difference of smoothness properties of eigenvectors [16

16. D.C. Dobson, “An efficient band structure calculations in 2D photonic crystals,” J. Comput. Phys. **149**, 363–376 (1999). [CrossRef]

21. L.W. Cordes and B. Moran, “Treatment of material discontinuity in the Element-Free Galerkin method,” Comput. Meth. Appl. Mech. Eng. **139**, 75–89 (1996). [CrossRef]

### 5.2 Square lattice of circular rods

*r*=0.2

*a*for this example. Dielectric constants are ε

_{m}=1.0 and ε

_{r}=8.9 for matrix and rods respectively. In this case, there exists a wide band gap in TM modes as widely known in literatures. Our results on band structures also confirm the wide band gap in TM modes as shown in Fig. 7. In this example, the width of band gap is determined by the 1st TM mode at point M and the 2nd TM mode at point X. The numerical results at these points obtained by both the MLS method and plane wave method are compared in Fig. 8, where the faster converging behavior of the method over the plane wave method are demonstrated. In the case of MLS method, the x-axis indicates the size of discretization such as

*h*=

*a*/(

*NP*-1), where

*NP*denotes the number of nodes in one direction. For the plane wave method, it similarly implies

*h*=

*a*/(

*N*

_{PW}-1), where

*N*

_{PW}means the number of plane wave used for the computation. This notation is arbitrary because the MLS method solves generalized eigenvalue problems, while the plane wave method does standard eigenvalue problems. The MLS method thus requires more computing time than the conventional plane wave method in general. Nonetheless, it is convenient for the purpose of direct comparison as in Fig. 8, because both (

*NP*)

^{2}and (

*N*

_{PW})

^{2}determine the size of system matrix to be solved.

## 6. Concluding remarks

14. L. Shen, S. He, and S. Xiao, “A finite-diference eigenvalue algorithm for calculating the band structure of a photonic crystal,” Comput. Phys. Comm. **143**, 213–221 (2002). [CrossRef]

16. D.C. Dobson, “An efficient band structure calculations in 2D photonic crystals,” J. Comput. Phys. **149**, 363–376 (1999). [CrossRef]

21. L.W. Cordes and B. Moran, “Treatment of material discontinuity in the Element-Free Galerkin method,” Comput. Meth. Appl. Mech. Eng. **139**, 75–89 (1996). [CrossRef]

## Appendix

20. D.W. Kim and Y. Kim, “Point collocation method using the fast moving least-square reproducing kernel approximation,” Int. J. Numer. Methods Engrg. **56**, 1445–1464 (2003). [CrossRef]

*u*(x) as

**p**(

**x**) must contain at least 1 and monomials in order to fulfill the linear reproducing conditions. For example,

**p**(

**x**)=[1,

*x, y, z*]

^{T}for 3D. By reproducing conditions, we mean that the MLS approximation can reproduce exactly the functions which appears in

**p**(

**x**). This is the well-known feature of MLS-based meshfree methods. Any polynomials of higher order or arbitrary types of functions can be added in the component of

**p**(

**x**) for the specific purpose of enrichment in MLS approximation.

*I*=1, …,

*NP*), the idea of moving least-square interpolant is used in determining the coefficient vector a(x̄). That is, a(¯x) is determined by minimizing the local error residual functional that is expressed as

*W*(x) is a compactly-supported continuous function. By solving ∂

*J*/∂a=0 for a(x̄), we have

*u*(x) as

*u*(x) is obtained by taking the limit of Eq. (20) as x̄→x,

**p**(0)=[1, 0, …, 0]

^{T}. The moment matrix

**M**(x) becomes

*u*

_{I}=

*u*(x

_{I}) and the MLS shape function

*N*

_{I}(x)=

*N*(x

_{I}-x) as

**M**(x), it is straightforward to verify that the shape function can also be written as

**55**, 1–34 (2002). [CrossRef]

20. D.W. Kim and Y. Kim, “Point collocation method using the fast moving least-square reproducing kernel approximation,” Int. J. Numer. Methods Engrg. **56**, 1445–1464 (2003). [CrossRef]

## Acknowledgements

## References and links

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

2. | Y. Xia, “Photonic crystals,” Adv. Mater. |

3. | K. Busch, “Photonic band structure theory: assesment and perspectives,” C. R. Physique |

4. | D. Cassagne, “Photonic band gap materials,” Ann. Phys. Fr. |

5. | J.B. Pendry, “Calculating photonic band structure,” J. Phys.: Condens. Matter |

6. | H.S. Sözüer, J.W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B |

7. | R.D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, and O.L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B |

8. | C.T. Chan, Q.L. Yu, and K.M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B |

9. | A.J. Ward and J.B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B |

10. | M. Qiu and S. 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. |

11. | K.M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B |

12. | X. Wang, X.G. Zhang, Q. Yu, and B.N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B |

13. | J.B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. |

14. | L. Shen, S. He, and S. Xiao, “A finite-diference eigenvalue algorithm for calculating the band structure of a photonic crystal,” Comput. Phys. Comm. |

15. | W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case,” J. Comput. Phys. |

16. | D.C. Dobson, “An efficient band structure calculations in 2D photonic crystals,” J. Comput. Phys. |

17. | C. Mias, J.P. Webb, and R.L. Ferrari, “Finite element modelling of electromagnetic waves in doubly and triply periodic structures,” IEE Proc.-Optoelectron. , |

18. | M. Marrone, V.F. Rodriguez-Esquerre, and H.E. Hernandez-Figueroa, “Novel numerical method for the analysis of 2D photonic crystals: the cell method,” Opt. Express |

19. | S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Review , |

20. | D.W. Kim and Y. Kim, “Point collocation method using the fast moving least-square reproducing kernel approximation,” Int. J. Numer. Methods Engrg. |

21. | L.W. Cordes and B. Moran, “Treatment of material discontinuity in the Element-Free Galerkin method,” Comput. Meth. Appl. Mech. Eng. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 6, 2003

Revised Manuscript: March 12, 2003

Published: March 24, 2003

**Citation**

Sukky Jun, Young-Sam Cho, and Seyoung Im, "Moving least-square method for the band-structure calculation of 2D photonic crystals," Opt. Express **11**, 541-551 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-6-541

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### References

- J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- Y. Xia, �??Photonic crystals,�?? Adv. Mater. 13, 369 (2001) and papers in this special issue. [CrossRef]
- K. Busch, "Photonic band structure theory: assesment and perspectives,�?? C. R. Physique 3, 53-66 (2002). [CrossRef]
- D. Cassagne, "Photonic band gap materials,�?? Ann. Phys. Fr. 23(4), 1-91 (1998). [CrossRef]
- J.B. Pendry, �??Calculating photonic band structure,�?? J. Phys.: Condens. Matter 8, 1085-1108 (1996). [CrossRef]
- H.S. Sozuer, J.W. Haus, and R. Inguva, �??Photonic bands: Convergence problems with the planewave method,�?? Phys. Rev. B 45, 13962-13972 (1992). [CrossRef]
- R.D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, O.L. Alerhand, �??Accurate theoretical analysis of photonic band-gap materials,�?? Phys. Rev. B 48, 8434-8437 (1993). [CrossRef]
- C.T. Chan, Q.L. Yu, and K.M. Ho, �??Order-N spectral method for electromagnetic waves,�?? Phys. Rev. B 51, 16635-16642 (1995). [CrossRef]
- A.J. Ward and J.B. Pendry, �??Calculating photonic Green�??s functions using a nonorthogonal finite difference time-domain method,�?? Phys. Rev. B 58, 7252-7259 (1998). [CrossRef]
- M. Qiu and S. 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]
- K.M. Leung and Y. Qiu, �??Multiple-scattering calculation of the two-dimensional photonic band structure,�?? Phys. Rev. B 48, 7767-7771 (1993). [CrossRef]
- X. Wang, X.G. Zhang, Q. Yu, and B.N. Harmon, �??Multiple-scattering theory for electromagnetic waves,�?? Phys. Rev. B 47, 4161-4167 (1993). [CrossRef]
- J.B. Pendry and A. MacKinnon, �??Calculation of photon dispersion relations,�?? Phys. Rev. Lett. 69, 2772-2775 (1992). [CrossRef] [PubMed]
- L. Shen, S. He, and S. Xiao, �??A finite-difference eigenvalue algorithm for calculating the band structure of a photonic crystal,�?? Comput. Phys. Comm. 143, 213-221 (2002). [CrossRef]
- W. Axmann and P. Kuchment, �??An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case,�?? J. Comput. Phys. 150, 468-481 (1999). [CrossRef]
- D.C. Dobson, �??An efficient band structure calculations in 2D photonic crystals,�?? J. Comput. Phys. 149, 363-376 (1999). [CrossRef]
- C. Mias, J.P. Webb and R.L. Ferrari, �??Finite element modelling of electromagnetic waves in doubly and triply periodic structures,�?? IEE Proc. Optoelectron. 146(2), 111-118 (1999). [CrossRef]
- M. Marrone, V.F. Rodriguez-Esquerre, and H.E. Hernandez-Figueroa, �??Novel numerical method for the analysis of 2D photonic crystals: the cell method,�?? Opt. Express 10, 1299-1304 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1299">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1299</a> [CrossRef] [PubMed]
- S. Li and W.K. Liu, �??Meshfree and particle methods and their applications,�?? Appl. Mechanics Rev. 55, 1-34 (2002). [CrossRef]
- D.W. Kim and Y. Kim, �??Point collocation method using the fast moving least-square reproducing kernel approximation,�?? Int. J. Numer. Methods Engrg. 56, 1445 - 1464 (2003). [CrossRef]
- L.W. Cordes and B. Moran, �??Treatment of material discontinuity in the Element-Free Galerkin method,�?? Comput. Meth. Appl. Mech. Eng. 139, 75-89 (1996). [CrossRef]

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