## Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm

Optics Express, Vol. 15, Issue 9, pp. 5416-5430 (2007)

http://dx.doi.org/10.1364/OE.15.005416

Acrobat PDF (367 KB)

### Abstract

A finite element method based eigenvalue algorithm is developed for the analysis of band structures of two-dimensional non-diagonal anisotropic photonic crystals under the in-plane wave propagation. The characteristics of band structures for the square and triangular lattices consisting of anisotropic materials are examined in detail and the intrinsic effect of anisotropy on the construction of band structures is investigated. We discover some interesting relationships of band structures for certain directions of the wave vector in the first Brillouin zone and present a theoretical explanation for this phenomenon. The complete band structures can be conveniently constructed by means of this concept.

© 2007 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

11. 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]

13. C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express **12**, 1397–1408 (2004). [CrossRef] [PubMed]

14. P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E **75**, 026703 (2007). [CrossRef]

15. G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and D. den Engelsen, “Symmetry properties of two-dimensional anisotropic photonic crystals,” J. Opt. Soc. Am. A **23**, 2002–2013 (2006). [CrossRef]

10. C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Phys. Rev. B **72**, 045133 (2005). [CrossRef]

## 2. Formulation

*t*) dependence of the form exp(

*jωt*) being implied, Maxwell’s curl equations can be expressed as

*μ*

_{0}and

*ε*

_{0}are the permeability and permittivity of free space, and [

*μ*] and [

_{r}*ε*] are, respectively, the relative permeability and permittivity tensors of the medium given by

_{r}*z*direction and periodic in the

*x*-

*y*plane, the wave modes in the PC for the in-plane propagation can be decoupled into transverse-electric (TE) and transverse-magnetic (TM) to

*z*modes if the tensor elements

*μ*and

_{mn}*ε*((

_{mn}*m*,

*n*) = (

*x*,

*z*), (

*y*,

*z*), (

*z*,

*x*), and (

*z*,

*y*)) are set to be zero.

### 2.1. The TE and TM modes

*H*,

_{z}*E*, and

_{x}*E*components, the above curl equations can be reduced to

_{y}*E*and

_{x}*E*in terms of

_{y}*H*. Substituting these expressions into Eq. (5), we can obtain the governing equation for the TE mode as

_{z}*k*

_{0}= ω√

*μ*

_{0}ε

_{0}is the wave vector in free space.

*E*,

_{z}*H*, and

_{x}*H*components, Maxwell’s curl equations can be simplified to

_{y}*H*and

_{x}*H*in terms of

_{y}*E*. Substituting these expressions into Eq. (11), again we achieve the governing equation for the TM mode as

_{z}*φ*=

_{z}*H*) and

_{z}*φ*=

_{z}*E*). Note that Eq. (13) applies to both magnetic and dielectric materials with non-diagonal permeability and permittivity tensors provided that the TE and TM modes are decoupled.

_{z}### 2.2. Finite element discretization

*H*or

_{z}*E*in Eq. (13), the quadratic triangular element [16,17

_{z}17. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. **18**, 737–743 (2000). [CrossRef]

_{z1}-φ

_{z6}, one at each of its three vertices and the other three at the middles of its three sides, as shown in Fig. 1(a), is extremely suitable for the discretization of the FEM. Moreover, in order to improve the accuracy and efficiency in the analysis, the curvilinear counterpart of Fig. 1(a), depicted in Fig. 1(b), is preferred and is adopted in our FEM based eigenvalue algorithm. The six shape functions of the quadratic triangular element are given by

*L*(

_{i}*i*= 1,2,3) are the simplex coordinates defined as

*x*and

_{i}*y*are the Cartesian coordinates of the

_{i}*i*th vertex and Δ represents the area of the triangular element, that is,

*i*in Eqs. (20)–(23) assumes values 1,2,3, cyclically, so that if

*i*= 3, then

*i*+1 = 1.

*x*and

*y*, can be approximated as

*x*and

_{j}*y*are the Cartesian coordinates of the

_{j}*j*th node (

*j*= 1–6) within each element and

*N*is defined in Eq. (18). Noting that

_{i}*L*

_{1}+

*L*

_{2}+

*L*

_{3}= 1 and selecting

*L*

_{1}and

*L*

_{2}as the independent variables, the relation of differentiation between the Cartesian coordinate system and the simplex coordinate system can be written as

*J*] is the Jacobian matrix of the transformation given by

*f*(

*x*,

*y*) in the Cartesian coordinate system can be performed in the simplex coordinate system through

*J*| is the Jacobian, the determinant of [

*J*]. Hence, the required computation for element matrices associated with the governing equation can be obtained directly in the simplex coordinate system.

*φ*in each element can be expanded as

_{z}*φ*= {

_{z}*N*}

*T*{

*φ*}, where {

^{e}_{z}*φ*} is the variable vector for each element, and

^{e}_{z}*T*denotes transpose. Applying Galerkin’s method and assembling all element matrices, Eq. (13) can be transformed into the matrix form as

*l*denotes the contour or boundary enclosing the area,

*n*̂ is its outward normal vector, and ∑

*extends over all different elements. Note that Eq. (29) is not an eigenvalue matrix equation due to the existence of the vector {*

_{e}*Ψ*} on the right-hand side. Fortunately, this term can be eliminated as a consequence of imposing the PBCs in our numerical model, as will be described in the next subsection.

### 2.3. Periodic boundary conditions for the 2D PCs

*φ*}

_{z}_{0}stands for the vector of variables in the interior region, the subscripts, I, II, III, and IV, denote the sides at which the variables locate, and

*K*] and [

*M*] in Eq. (39) can be combined through Eqs. (33) and (36), respectively. Furthermore, if we divide row 2 in Eq. (39) by

*e*-

*and add it to row 4, {*

^{Jkxa}*Ψ*}

_{I}and {

*Ψ*}

_{III}will cancel each other by the use of Eqs. (34) and (35). Similarly, {

*Ψ*}

_{II}and {

*Ψ*}

_{IV}can cancel each other when row 3 in Eq. (39) is divided by

*e*-

*and then added to row 5 with the help of Eqs. (37) and (38). By means of these operations, Eq. (39) becomes*

^{Jkxa}*a*from the other two vertices. In each part, the fields at the three vertices can be linked together using Eqs. (45)–(53). The arrows in Fig. 2(b) illustrate this idea. Same as in the case of square unit cell, associating the information at sides I, II, and III with that at sides IV, V, and VI, respectively, will suitably modify the matrices and vectors in Eq. (29), and consequently the desired eigenvalue matrix equation can be derived.

## 3. Numerical results

*ϵ*] of the nematic LCs we use are given as

_{r}*n*= 1.5292 and

_{o}*n*= 1.7072 are, respectively, the ordinary and extraordinary refractive indices of the nematic LCs,

_{e}*θ*is the angle between the crystal

_{c}*c*-axis and the

*z*-axis, and

*φ*represents the angle between the projection of the crystal

_{c}*c*-axis on the

*x*-

*y*plane and the

*x*-axis, as defined in Fig. 3. For the decoupling of the TE and TM modes in the 2D condition. The angle,

*θ*, is set to be 90° and thus the LC molecules will merely rotate in the

_{c}*x*-

*y*plane.

*z*-component and will regard the anisotropic PC as an isotropic one. So we will focus our discussion on the band structures of the TE mode. From the numerical results we will explain the concept of constructing the band structures for anisotropic PCs clearly.

### 3.1. Square lattice

*ε*= 11.56 and radius

_{r}*r*= 0.2

*a*in the background material of LCs. The unit cell of this structure is also indicated by the square region in Fig. 4(a), and the corresponding first BZ of the reciprocal lattice is depicted in Fig. 4(b). For isotropic PCs, a complete band structure can be conveniently constructed by simply considering all the possible directions of the wave vector

**k**restricted in the irreducible BZ (IBZ) which is enclosed by points Γ, X, and M, as shown in Fig. 4(b). The reliability and completeness of the band structure obtained from this IBZ for isotropic PCs are fully based on the structure symmetry, including the rotation and reflection symmetry. For anisotropic PCs, the situation becomes more complicated because the circumstance experienced by the EM waves is highly direction-dependent. The anisotropy may break the structure symmetry, which in isotropic cases can be totally determined by the spatial configuration of materials. With this consideration, it is reasonable to inspect more band structures constructed from different directions of k in the first BZ.

*φ*= 30° also prevents the reflection symmetry from a reasonable explanation. To explain what causes the identity of segments X-M and X′′-M′, we appeal to the basic principle of periodic structures. For periodic structures, every lattice vector

_{c}**R**and reciprocal lattice vector

**G**must satisfy the requirement

**G**∙

**R**= 2

*πN*where

*N*is an integer [3]. Thus if

**k**is incremented by

**G**, the phase between cells is incremented by

**G**∙

**R**, which is equal to 2

*πN*and does not really affect the mode. For every pair of wave vectors,

**k**and

_{1}**k**, of segments X-M and X′′-M′, the difference between them is exactly a reciprocal lattice vector

_{2}**G**, as depicted in Fig. 6(b), so it is no surprise that the band structures of these two segments are the same. Next we examine segments X′-M and X′-M′, another identical pair as shown in Fig. 5(d). The identity of this pair can be reasonably accepted by applying the two reasons just mentioned above simultaneously. This idea is illustrated in Fig. 6(c) which reveals that the difference between

**k**and

_{1}**k**is also exactly

_{2}**G**. In Fig. 5(b) and (e), no conditions mentioned above occur, and, consequently, the whole band structures from these corresponding sub-zones are quite unlike.

*φ*= 45°, another interesting behavior of the band structure can be observed. With LC molecules orienting to this special direction, the resultant structure, determined by the arrangement of silicon rods and the orientation of LC molecules, has reflection symmetry about the

_{c}*c*-axis of LCs. According to this observation, it is absolutely reasonable to predict that the band structures from Γ-X-M and Γ-X′-M′ are identical to those from Γ-X′-M and Γ-X′′-M′, respectively. The reflection symmetry of structures will vary with the rotation angle of LC molecules, but the primary idea discussed above works well and can be considered as a general principle applied to any anisotropic PCs.

*φ*= 0°, 30°, and 45° in Fig. 7 from which we can see that there are no photonic band gaps for these different LC rotation angles. Notice that these band structures are constructed from the four sub-zones and the shadowed regions represent ignorable parts because they can be duplicated from unshadowed parts.

_{c}### 3.2. Triangular lattice

*φ*= 0°, 30°, and 45° in Fig. 9. To get an insight into the effect of the anisotropic material on the band gap, we concentrate on Fig. 9(a), the case of

_{c}*φ*= 0°, first. The red region, if exists, in every one of the six distinct sub-zones stands for the band gap of the structure found from the corresponding sub-zone. For

_{c}*φ*= 0°, Γ-M-K, Γ-M-K′, and Γ-M′-K′ are identical to Γ-M′′-K′′, Γ-M′′-K′′ , and Γ-M′-K′′, respectively. Accordingly, it is obvious that the band gaps corresponding to these respective pairs are exactly the same to each other. The main point worth emphasizing here is that the normalized frequencies between which the band gaps exist for Γ-M-K, Γ-M-K′, and Γ-M′-K′ are different. So to describe the band gap of this structure suitably, we should find the normalized frequency range appearing simultaneously in all distinct band gaps. Then this normalized frequent range will stand for the correct band gap of the structure. Specifically, the band gap found from Γ-M-K falls between the normalized frequencies of 0.636 and 0.656, whereas the two numbers obtained from Γ-M-K′ and Γ-M′-K′ are, respectively, 0.635 and 0.640, and 0.634 and 0.639. As a result, the correct band gap of this PC for

_{c}*φ*= 0° will exist when the normalized frequency falls between 0.636 and 0.639. In Fig. 9(b) and (c), there are two parts, obtained from Γ-M-K′ and Γ-M′-K′, that do not have band gaps, and hence there do not exist band gaps at all directions for

_{c}*φ*= 30° and

_{c}*φ*= 45°. With these examples, we can see that the anisotropy indeed has a great effect on the behavior of band structures and should be taken into account for constructing a complete band structure.

_{c}## 4. Conclusion

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

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

4. | J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. |

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

6. | P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B |

7. | H. A. Haus, |

8. | I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B |

9. | Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. |

10. | C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Phys. Rev. B |

11. | 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. |

12. | L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, “An efficient finite-element method for the analysis of photonic band-gap materials,” in 1999 IEEE MTT-S Dig. |

13. | C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express |

14. | P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E |

15. | G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and D. den Engelsen, “Symmetry properties of two-dimensional anisotropic photonic crystals,” J. Opt. Soc. Am. A |

16. | J. Jin, |

17. | M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. |

18. | P. Yeh and C. Gu, |

**OCIS Codes**

(160.3710) Materials : Liquid crystals

(230.3990) Optical devices : Micro-optical devices

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: January 22, 2007

Revised Manuscript: April 14, 2007

Manuscript Accepted: April 17, 2007

Published: April 19, 2007

**Citation**

Sen-ming Hsu, Ming-mung Chen, and Hung-chun Chang, "Investigation of band structures for 2D non-diagonal anisotropic photonic
crystals using a finite element method based eigenvalue algorithm," Opt. Express **15**, 5416-5430 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5416

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### References

- E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
- S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, N. J., 1995).
- J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
- S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999). [CrossRef]
- P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996). [CrossRef]
- H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1984).
- I. H. H. Zabel and D. Stroud, "Photonic band structures of optically anisotropic periodic arrays," Phys. Rev. B 48, 5004-5012 (1993). [CrossRef]
- Z. Y. Li, B. Y. Gu, and G. Z. Yang, "Large absolute band gap in 2D anisotropic photonic crystals," Phys. Rev. Lett. 81, 2574-2577 (1998). [CrossRef]
- C. Y. Liu and L. W. Chen, "Tunable band gap in a photonic crystal modulated by a nematic liquid crystal," Phys. Rev. B 72, 045133 (2005). [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]
- L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, "An efficient finite-element method for the analysis of photonic band-gap materials," in 1999 IEEE MTT-S Dig. 4, 1703-1706 (1999).
- C. P. Yu and H. C. Chang, "Compact finite-difference frequency-domain method for the analysis of twodimensional photonic crystals," Opt. Express 12, 1397-1408 (2004). [CrossRef] [PubMed]
- P. J. Chiang, C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007). [CrossRef]
- G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and D. den Engelsen, "Symmetry properties of two-dimensional anisotropic photonic crystals," J. Opt. Soc. Am. A 23, 2002-2013 (2006). [CrossRef]
- J. Jin, Finite Element Method in Electromagnetics (John Wiley and Sons, Inc., New York, 2002).
- M. Koshiba and Y. Tsuji, "Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems," J. Lightwave Technol. 18, 737-743 (2000). [CrossRef]
- P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons, Inc., New York, 1999).

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