## Novel numerical method for the analysis of 2D photonic crystals: the cell method

Optics Express, Vol. 10, Issue 22, pp. 1299-1304 (2002)

http://dx.doi.org/10.1364/OE.10.001299

Acrobat PDF (422 KB)

### Abstract

The Cell Method, a new efficient numerical method suitable for working with periodic structures having anisotropic inhomogeneous media with curved shapes, is proposed in order to calculate the band gap of 2D photonic crystals for in-plane propagation of TM and TE waves. Moreover some numerical comparisons with other numerical methods will be provided.

© 2002 Optical Society of America

## 1. Introduction

2. H.Y.D. Yang,”Finite Difference Analysis of 2-D Photonic Crystals,” IEEE Trans. Microwave Theory Tech. **44**2688–2695 (1996). [CrossRef]

## 2. The Cell Method

- Two kinds of oriented space elements (space cells): the space elements endowed with an inner orientation that we will call primal space cells (primal points
**P**, lines**L**, surfaces**S**and volumes**V**) and the space elements endowed with an outer orientation that we will call dual space cells (dual points**P̃**, lines*L̃*, surfaces*S̃*and volumes*Ṽ*) (Fig.1a). - Two oriented cell complexes and a relationship of duality between them. A space complex, synonymous of three dimensional grid, is a structured collection of points, lines, surfaces and volumes (cells).The primal complex is made by inner oriented space cells, the dual complex is made by outer oriented space cells and the relationship of duality means that we need a biunivocal matching respectively between the primal pointsFig. 1.(a)Inner and outer orientations of the space cells in the 3D space (b)Primal and dual complexes and global variables associated with their primal and dual space cells respectively.
*P*and the dual volumes*Ṽ*, the primal lines*L*and the dual surfaces*S̃*, the primal surfaces**S**and the dual lines*L̃*and between the primal volumes**V**and the dual points**P̃**. - The use of the global (integral) variables to represent physical variables and their physically coherent association with the oriented space cells (Fig.1b)[3][4]:Electric voltage
*V*[**L**](=*∫*E·_{L}*d***l**), Magnetic flux Φ[**S**](=*∫*_{S}**B**·*d***s**), Magnetic voltage*F*[**L̃**](=*∫*_{L̃}**H**·*d***l**), Electric flux ψ[**S̃**](=*∫*_{S̃}**D**·*d***s**), Electric charge content*Q*[_{c}**Ṽ**](=*∫*), Electric current_{Ṽ}ρdv*I*[**S̃**](=*∫*_{S̃}**J**·*d*s).The electromagnetic laws, that are links between global variables, can be divided into two categories, according to the type of variables that are linked: *Topological equations*(Field equations).These equations link global variables associated with space cells belonging to the same kind of complex (either the primal or the dual one).Since they do not concern metric concepts, they can be enforced on the cell complexes in a discrete not approximated form by using appropriate incidence matrices. If we denote with**G**,**C**,**D**and**G̃**=**D**,^{T}**C̃**=**C**,^{T}**D̃**= -**G**the incidence matrices [7] related to the primal and dual cell complexes respectively, which are the discrete counterparts of the differential operators^{T}*gradient, curl*and*divergence*, the topological equations in the frequency domain can be expressed as follows:*Faraday-Neumann law***Ca**= -jωΦ*Magnetic Gauss law***DΦ**= 0*Maxwell-Ampére law***C̃F**=*jω***ψ**+**I***Electric Gauss law***D̃ψ**=**Q**_{c}

where**V, Φ, F, ψ, I, Q**_{c}are scalar arrays.*Constitutive relations*.These equations link global variables associated with space cells belonging to different kind of cell complexes (i.e. the electric constitutive relation links the electric voltage*V*, associated with the primal lines**L**, and the electric flux ψ, associated with the dual surfaces**S̃**).They concern metric concepts and material parameters and they can be enforced on the cell complexes only approximatively in a discrete form by using suitable constitutive matrices:If an orthogonality occurs (Fig. 2.Examples of 2D photonic crystals and periodicity of the unit cells in the xy-plane (a) for rectangular unit cells (b) for quadrangular unit cells. (c) Dependence of some voltages on the boundary of a rectangular unit cell by the periodic boundary condition.**L**⊥**S̃**and**L̃**⊥**S**, as in the Fig.1b), then the matrices**M**_{ε},M_{μ}can be built diagonal [7] and assure, at the same time, the convergence of the method. In general, both for orthogonal or non-orthogonal grids, it is possible to build the matrices**M**_{ε},**M**_{μ}by the*Microcell Interpolation Scheme*(MIS) [4].

## 3. Analysis of 2d periodic structures with the Cell Method

2. H.Y.D. Yang,”Finite Difference Analysis of 2-D Photonic Crystals,” IEEE Trans. Microwave Theory Tech. **44**2688–2695 (1996). [CrossRef]

### 3.1 Two dimensional TMz case

*z*-axes (Fig.3). This fact allows us to work like in a pure 2D case (Fig.4a) by projecting the 3D cell complexes on the xy-plane, where:

- The 2D unit cell is discretized by a triangular primal cell complex and a barycentric dual cell complex (Fig.4a).
- The association global variables → space cells is:Electric voltage
*V*→ Primal point**P**, Magnetic flux Φ → Primal line**L**Magnetic voltage*F*→ Dual line**L̃**, Electric flux ψ → Dual surface**S̃** - The topological equations are:Faraday-Neumann law:
**GV**= -*jω*Φ, Maxwell-Ampére law:**C̃F**=*jω***ψ**. - The constitutive relations are:Electric:
**ψ**=**M**_{ε}**V**, Magnetic:**F**=**M**_{ν}**Φ**.

**C̃**= -

**G**

^{T}in the 2

*D*cases, we obtain the matricial equation:

**V**is the array of the

*N*(=number of nodes=number of primal points P) electric voltages. If we build the matrices

**M**

_{ν}and

**M**

_{ε}with the MIS, for triangular grids the

**G**

^{T}

**M**

_{ν}

**G**is a symmetric positive semidefinite band diagonal matrix and the

**M**

_{ε}is a diagonal positive definite matrix. For a rectangular unit cell of sizes [

*a, b*] (Fig.2a) the appropriate boundary condition is [2

2. H.Y.D. Yang,”Finite Difference Analysis of 2-D Photonic Crystals,” IEEE Trans. Microwave Theory Tech. **44**2688–2695 (1996). [CrossRef]

*β*and

_{x}*β*are the phase constants in the

_{y}*x*and

*y*directions. Since some voltages on the boundary are dependent of other voltages, because the periodic conditions (Fig.2c), it is possible to reduce the dimension of the problem by means of the following relation:

**V**′ is a vector of the independent voltages,

**T**(

*β*,

_{x}*β*) is a sparse matrix whose entries

_{y}**T**

_{ij}are 1 if

**V**

_{i}=

**V**′

_{j}, either

*e*

^{-jβxa}or

*e*

^{-jβyb}or

*e*

^{-jβxa-jβyb}according to the boundary condition between

**V**

_{i}and

**V**′

_{j}, 0 otherwise. Thus the Helmholtz equation with the periodic boundary conditions reduces to:

**A**=

**T**

^{H}

**G**

^{T}

**M**

_{ν}

**GT**is a hermitian semidefinite positive matrix and

**B**=

**T**

^{H}

**M**

_{ε}

**T**is a diagonal real positive definite matrix. This generalized eigenvalues problem can be transformed into a simple eigenvalues problem:

**D**= √

**B**,

**M**

_{TM}=

**D**

^{-1}

**AD**

^{-1}and

**V**’’ =

**DV**′.The matrix

**M**

_{TM}is still sparse hermitian positive semidefinite and, by an appropriate renumbering of the nodes, the structure of this matrix becomes quasi band diagonal. Thus, in order to find its eigenvalues, can be used the same efficient method developed for the FD method in [2

**44**2688–2695 (1996). [CrossRef]

### 3.2 Two dimensional TEz case

- The 2D unit cell is discretized by a barycentric primal cell complex and a triangular dual cell complex.
- The association global variables → space cells is:Electric voltage
*V*→ Primal line**L**, Magnetic flux Φ → Primal surface**S**, Magnetic voltage*F*→ Dual point*P̃*, Electric flux ψ → Dual line**L̃**. - The topological equations are:Maxwell-Ampére law:
**G̃F**=*jω***ψ**, Faraday-Neumann law:**Ca**= -*jω***Φ**. - The constitutive relations are:Electric:
**V**=**M**_{ε-1}**ψ**, Magnetic:**Φ**=**M**_{μ}**F**.

*G̃*

^{T}=

**C**in the 2

*D*cases, we obtain the matricial equation:

**F**is the array of the

*N*(=number of nodes=number of dual points

*P̃*) magnetic voltages. For a rectangular unit cell of sizes [

*a, b*], by the boundary condition:

## 4. Numerical results and discussion

*TM*and

*TE*waves in a square lattice (lattice constant

*a*) of dielectric rods (

*∊*= 12, radius=0.2a). The results, compared with PWE [5

_{r}5. S.G. Johnson, S. Fan, P.R. Villeneuve, J.D. Joannopulos, and L.A. Kolodziejski, ”Guided modes in photonic crystal slabs,” Phys. Rev. B **3**, 5751–5758 (1999) [CrossRef]

*TM*photonic band gap (Fig.5a). In (Fig.5b) the normalized frequencies of the first

*TM*mode in

*M*and the second

*TM*mode in

*X*, for the first test, calculated by CM and FEM, are compared for different number of grid points. The second test was the calculus of the photonic band structure for

*TM*and

*TE*waves in a triangular lattice (lattice constant

*a*) of air columns (radius=0.48a) in a dielectric substrate (

*∊*= 13).The results, in excellent agreement with [1] where a PWE has been used, exhibit a complete band gap (Fig.6a). The third test was the calculus of the photonic band structure for

_{r}*TM*and

*TE*waves in a square lattice of anisotropic dielectric rods (

*∊*=

_{xx}*∊*= 23.04,

_{yy}*∊*= 38.44, Filling Fraction=0.4) (Fig.6b). The results, in good agreement with [6] where a PWE has been used, exhibit a complete band gap between (0.219–0.254) 2

_{zz}*πc*/

*a*.In conclusion, the good agreement of the CM results with the results of other methods shows the effectiveness of the present approach. Moreover, in the test 1, CM exhibits more accuracy than FEM for the same number of grid points.

## 5. Conclusion

## References and links

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

2. | H.Y.D. Yang,”Finite Difference Analysis of 2-D Photonic Crystals,” IEEE Trans. Microwave Theory Tech. |

3. | E. Tonti, ”Finite Formulation of the Electromagnetic Field,” in |

4. | M. Marrone, ”Computational Aspects of Cell Method in Electrodynamics,” in |

5. | S.G. Johnson, S. Fan, P.R. Villeneuve, J.D. Joannopulos, and L.A. Kolodziejski, ”Guided modes in photonic crystal slabs,” Phys. Rev. B |

6. | Z.Y. Li, B.Y. Gu, and G.Z. Yang, ”Improvement of absolute band gaps in 2Dphotonic crystals by anisotropy in dielectricity,” Eur.Phys. J. B |

7. | M. Clemens and T. Weiland, ”Discrete Electromagnetism with the Finite Integration Technique,” in |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(160.1190) Materials : Anisotropic optical materials

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 30, 2002

Revised Manuscript: October 21, 2002

Published: November 4, 2002

**Citation**

Massimiliano Marrone, V. Rodriguez-Esquerre, and H. Hernandez-Figueroa, "Novel numerical method for the analysis of 2D photonic crystals: the cell method," Opt. Express **10**, 1299-1304 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-22-1299

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### References

- J. D. Joannopulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton University Press, 1995).
- H. Y. D. Yang, "Finite Di.erence Analysis of 2-D Photonic Crystals," IEEE Trans. Microwave Theory Tech. 44 2688-2695 (1996). [CrossRef]
- E. Tonti, "Finite Formulation of the Electromagnetic Field," in Geometric Methods for Computational Electromagnetics, PIER 32, F. L.Teixeira, J. A.Kong, ed.(EMW Publishing 2001) 1-44.
- M. Marrone, �??Computational Aspects of Cell Method in Electrodynamics,�?? in Geometric Methods for Computational Electromagnetics, PIER 32, F. L.Teixeira, J. A. Kong, ed. (EMW Publishing 2001), 317-356.
- S. G. Johnson, S. Fan, P.R. Villeneuve, J. D. Joannopulos, L. A. Kolodziejski, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B 3, 5751-5758 (1999) [CrossRef]
- Z. Y. Li, B. Y. Gu, G. Z. Yang, �??Improvement of absolute band gaps in 2D photonic crystals by anisotropy in dielectricity,�?? Eur. Phys. J. B 11, 65-73 (1999).
- M. Clemens, T. Weiland, �??Discrete Electromagnetism with the Finite Integration Technique,�?? in Geometric Methods for Computational Electromagnetics, PIER 32, F. L. Teixeira, J. A. Kong, ed. (EMW Publishing 2001), 65-87.

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