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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 20 — Sep. 26, 2011
  • pp: 19346–19353
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The optimal structure of two dimensional photonic crystals with the large absolute band gap

Donglin Wang, Zhongyuan Yu, Yumin Liu, Pengfei Lu, Lihong Han, Hao Feng, Xiaotao Guo, and Han Ye  »View Author Affiliations


Optics Express, Vol. 19, Issue 20, pp. 19346-19353 (2011)
http://dx.doi.org/10.1364/OE.19.019346


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Abstract

This paper reports a new designed square lattice GaAs structure of two-dimensional photonic crystals with absolute band gap approach to 0.1623 (2πc/a), where a is the period of the square lattice. The optimal structure is obtained by combining the Geometry Projection Method and Finite Element Method. Both gradient information and symmetric control points are introduced to reduce the calculation cost. For benefit to the fabrication in reality, the structure is simplified by the combination of triangle and rectangular geometry. Through parameter optimization, the absolute band gap of the new structure is improved to 0.1735 (2πc/a), which is much larger than those reported before. The new PC structure is convenient and stab for fabrication, and may be found applications in the future optical devices.

© 2011 OSA

1. Introduction

Photonic Crystals (PCs) are structures with periodic dielectric materials distribution, which could inhibit the propagation of the plane polarized light at a certain frequency [1

1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]

]. The dielectric materials distribution of the PCs in the unit cell has a strong effect on the band gap property. The design of the band gap is very significant for the application of the PCs. Especially a large band gap size is very useful to control the wide band signals. In recent years, some novel PC structures with large absolute band gap have been reported. L. F. Shen has found that the structure with square holes and rods in square unit cell has the absolute band gaps of 0.1158(2πc/a) and 0.11(2πc/a) respectively [2

2. L. F. Shen, S. He, and S. S. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B 66(16), 165315 (2002). [CrossRef]

]. H. P. Li has reported that the large absolute band gap of 0.1466 (2πc/a) can be get by modifying the radius and positions of the dielectric rods [3

3. H. P. Li, L. Y. Jiang, W. Jia, H. X. Qiang, and X. Y. Li, “Genetic optimization of two-dimensional photonic crystals for large absolute band-gap,” Opt. Commun. 282(14), 3012–3017 (2009). [CrossRef]

]. S. Zarei has obtained an absolute band gap of 0.1522 (2πc/a) by reducing the symmetry of the PC structure [4

4. S. Zarei, M. Shahabadi, and S. Mohajerzadeh, “Symmetry reduction for maximization of higher-order stop-bands in two-dimensional photonic crystals,” J. Mod. Opt. 55(18), 2971–2980 (2008). [CrossRef]

]. And L. F. Shen has also found a structure with an absolute band gap of 0.157 (2πc/a) by using a novel optimization algorithms [5

5. L. F. Shen, Z. Ye, and S. L. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B 68(3), 035109 (2003). [CrossRef]

]. As the development of computer technique and calculation procedure, it is expect that a novel treatment processes could be helpful to design 2D PCs with a maximal absolute band gap.

Optimization design is an efficient way to find the optimal structure of the PC with maximal absolute band gap. Several Parameter optimization methods [6

6. M. Qiu and S. He, “Optimal design of a two-dimensional photonic crystal of square lattice with a large complete two-dimensional band gap,” J. Opt. Soc. Am. B 17(6), 1027–1030 (2000). [CrossRef]

8

8. F. Wen, S. David, X. Checoury, M. El Kurdi, and P. Boucaud, “Two-dimensional photonic crystals with large complete photonic band gaps in both TE and TM polarizations,” Opt. Express 16(16), 12278–12289 (2008). [CrossRef] [PubMed]

] have demonstrated the capacity to get the large absolute band gaps, but the global optimal solution is hard to reach because of trapping in to the multiple localized minimum values. Topology optimization method has the largest potential to find the global optimal solution, and it have been combined with several methods to optimize the structure of the PC for the maximum band gap, such as genetic algorithms [5

5. L. F. Shen, Z. Ye, and S. L. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B 68(3), 035109 (2003). [CrossRef]

], centroidal voronoi tessellation algorithm [9

9. O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett. 100(15), 153904 (2008). [CrossRef] [PubMed]

], convex conic optimization algorithm [10

10. H. Men, N. C. Nguyen, R. M. Freund, K. M. Lim, P. A. Parrilo, and J. Peraire, “Design of photonic crystals with multiple and combined band gaps,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(4), 046703 (2011). [CrossRef] [PubMed]

] and so on. Those methods divide the design domain into small pixels and scan all the possible configurations. Although some novel treatments can improve computational speed, the large amount of variables is still a major drawback of those methods. So a more efficient and novel method is still significant for approaching the global optimal structure of the PC.

2. Method

The highlight of our method is the reduction of the calculated cost by employing the GPM. This alternative approach for structure optimization avoids some weak points in standard topology algorithm, such as, intermediate dielectric parameter and checkerboard pattern. Although, filters and penalization algorithms have been successfully used to resolve these problems [14

14. O. Sigmund and J. Petersson, “Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima,” Struct. Optim. 16(1), 68–75 (1998). [CrossRef]

], the large calculation quantity is still an important problem to be solved urgently. And the GPM is suitable to be used in this field for the PC structure optimization [15

15. W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett. 32(1), 77–79 (2007). [CrossRef] [PubMed]

]. The details of the optimization method in the present paper are described as follows.

In step 1, we construct an initial distribution of the dielectric material by GPM. The initial structure can be arbitrary shape for opening a new gap, and it should have an existing gap for improving. We choose a structure with an absolute band gap of 0.0356 (2πc/a) as an example (as shown in Fig. 2
Fig. 2 The band structure of the initial structure, the solid lines are for the H polarization (TM) and the dashed lines for the E polarization (TE).
), and one-eighth of the structure is constructed as the initial distribution by GPM. The GPM is easy to be understood by following procedures [11

11. J. Norato, R. Haber, D. Tortorelli, and M. P. Bendsoe, “A geometry projection method for shape optimization,” Int. J. Numer. Methods Eng. 60(14), 2289–2312 (2004). [CrossRef]

]. First, we introduce a set of points at the design region, and give each control point a different height. In current study, one quarter of the unit structure is consider as the design region, 20 × 20 control points are arranged at that region. One half of the region is selected as the computational domain due to the symmetry of the structure, which means only 210 control points are used in the computation. Second, we fit a three-dimensional surface through the heights of control points. The surface function is given by
S(x)=c0+cx+pλpΦ(|xxp|),
(1)
Φ(r)=r2logr,
(2)
where x and xp are the coordinates of the refined mesh of the design domain (100 × 100 grids are considered in our study) and the mesh of the control points respectively. And the details of solving the coefficients c, c0 and λp are given in [16

16. G. Turk and J. F. O’Brien, “Modeling with Implicit Surfaces that Interpolate,” ACM Trans. Graph. 21(4), 855–873 (2002). [CrossRef]

]. The shape of the surface is controlled by the heights of the control points. It is the realization of the control on the refined mesh by fewer variables. In the final, we find the intersection curve of a level plane with the surface, and use the curve to define the interface between different dielectric material regions in two dimensions. As shown in Fig. 3
Fig. 3 The schema of the GPM, the distribution of the dielectric can be controlled by control points.
, the distribution of the dielectric materials can be written as
εr(x)=εr,min+(εr,maxεr,min)2(tanh[sign[S(x)]d(x)ξ]+1),
(3)
d(x)=minx|xx0|,
(4)
where ξ is used to control the intermediate dielectric at the interface between different dielectric material regions, and we can set available ξ to make the intermediate dielectric vanish (ξ = 100 in this paper). x0 is the coordinate of the intersection curve, and d(x) is minimum distance from any point to the nearest intersection curve. So we can construct the initial distribution of the dielectric material by giving the special heights to the control points, and adjusting those heights will change the distribution.

In step 2, we introduce the gradient information to reduce the design variables. The gradient is very difficult to get in the process of PC band gap optimization, but it will observably speed up the computational program. The reasonable utilization of the gradient information is very useful to increase computational efficiency. In this work, the sensitivity of the absolute band gap to the initial dielectric distribution of the 2D PC can be simply described asG=dφdε|ε=ε0, whereε0is the initial dielectric distribution andφ=wn+1wndenotes the absolute band gap (n = 5 for TE wave and n = 9 for TM wave in current work). The sensitivity distribution is shown in Fig. 4
Fig. 4 The sensitivity information of the absolute band gap to the initial dielectric structure. Left is the initial dielectric distribution of the 2D PC. Right is the sensitivity distribution, red represents the positive value, blue is the negative value.
, the positive value means that increasing the dielectric material distribution at that region is energetic to improve the band gap, and the negative region show the contrary. So we only design the control points which located at the positive domain for increasing the band gap. This treatment will reduce the variables from 210 (mentioned in step 1) to about 100. The calculation precision and efficiency still can meet demand, even though the variables are very less.

In step 3, we build up the data interface between the GPM and the FEM, and use FEM to compute the band gap property of the 2D PC. First, we adopt GMP to give a dielectric distribution of the PC and save the data in a special format which can be recognized by FEM, and then we use FEM to read data and reconstruct the structure of the PC in FEM. FEM is proved to be a useful tool for solving PC problems, such as H. D. Tian [17

17. H. Tian, Z. Yu, L. Han, and Y. Liu, “Birefringence and confinement loss properties in photonic crystal fibers under lateral stress,” IEEE Photon. Technol. Lett. 20(22), 1830–1832 (2008). [CrossRef]

,18

18. T. Hong-Da, Y. Zhong-Yuan, H. Li-Hong, and L. Yu-Min, “Lateral stress-induced propagation characteristics in photonic crystal fibres,” Chin. Phys. B 18(3), 1109–1115 (2009). [CrossRef]

] use FEM to compute PC fiber problem and J. S. Jensena use FEM to optimize PC waveguide bends [19

19. J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett. 84(12), 2022–2024 (2004). [CrossRef]

]. And the FEM can easily compute the band gap property of the 2D PC with arbitrary distribution. The ultrafine mesh is employed in the FEM for giving a precise result.

In step 4, we insert the design variables and the target function into a numerical optimization algorithm to find the optimal solution. In our optimization problem, the design variables are the heights of the control points, and the target function is the width of the absolute band gap. The problem can be solved by any no-liner programming algorithm. The Nelder Mead algorithm can effectively solve the problems of the unconstrained extreme value of the function with many variables, and it is employed here to find the optimal structure of the 2D PC.

3. Application

The calculation efficiency of the method is enhanced by introducing the symmetry of the structure and gradient information. The unit cell with 200 × 200 grids can be adjusted by about 100 control points. After optimization, the height of the control points is appropriate to project a new PC structure (as shown in Fig. 5
Fig. 5 The new structure with an absolute band gap of 0.1623 (2πc/a) is found by our method. Blue is the dielectric material distribution, and white indicates air.
) which has an absolute band gap of 0.1623 (2πc/a) at a midfrequency of 0.8690 (2πc/a).The absolute band gap of the new structure (as shown in Fig. 6
Fig. 6 The band structure of the new PC structure.
) is about 4.6 times larger than that for the initial structure. The method demonstrates a great capacity in improving an existing absolute band gap. The 1681 plane waves are employed in the plane wave expansion method to verify the optimization result, and it shows that the relative error is about 10−4.

The structure of Fig. 5 is probably not the global optimal solution because the method is strong correlated with the initial structure, but it promises a new PC geometry with larger absolute band gap. The parameter optimization of this new structure is very significant to achieve a larger absolute band gap and provide a facility in practical manufacturing. The parameterized structure is obtained directly by removing the insignificant dielectric distribution. As shown in Fig. 7
Fig. 7 The parameterized structure of the new PC structure, 12 geometric parameters are used to define the simplified structure.
, it's simplified by a structure which simply consists of triangle and rectangular. There are 12 geometric parameters (r 1, w 1, h 1, k 1, b 1, θ 1, r 2, w 2, h 2, k 2, b 2 and θ 2) are used to define the simplified structure. The initial values for these parameters are estimated by the structure in Fig. 5. After parameter optimization, a larger absolute band gap of 0.1735 (2πc/a) at a midfrequency of 0.8828 (2πc/a) and the relative band gap of 19.66% is found. The band structure of this optimal 2D PC is shown in Fig. 8
Fig. 8 The band structure of the optimal 2D PC structure with optimal parameters: r1 = 0.2918, w1 = 0.0284, h1 = 0.0501, k1 = 0.0656, b1 = 0.0203, θ1 = 48.6597°, r2 = 0.3125, w2 = 0.0228, h2 = 0.1203, k2 = 0.1164, b2 = 0.0447 and θ2 = 52.8430°.
. The optimal parameters are r 1 = 0.2918, w 1 = 0.0284, h 1 = 0.0501, k 1 = 0.0656, b 1 = 0.0203, θ 1 = 48.6597°, r 2 = 0.3125, w 2 = 0.0228, h 2 = 0.1203, k 2 = 0.1164, b 2 = 0.0447 and θ 2 = 52.8430°. The value of above parameters are normalized by lattice constant a = 1. In the applications, the lattice constant should scale down to a = 0.8828λ, where 0.8828 is the normalized midfrequency of the band gap and λ should be consistent with communication wavelength 1.55nm. So the precision of the optimal parameters should be 10−4 to meet the manufacturing precision. In practical fabrications, slightly dilated and eroded realizations are unavoidable, which will influence the stability of band structure. To confirm the stability of the optimal structure, we assume θ 1 = θ 2 = 50° and find that the change of the band gap is just 0.0005 (2πc/a). For providing a more effective result to experiments, we introduce the numerical error ±103and angle error±3°in the stability analysis, and find that the maximum error of the band gap is about 4%, which is stable for fabrication.

The optimal structure in our work can be simply split into two independent structure, a frame construction and eight dielectric rods. The similar frame construction and distribution of dielectric rods have been proved to be the structure with large band gap for TE and TM wave respectively [9

9. O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett. 100(15), 153904 (2008). [CrossRef] [PubMed]

]. The large absolute band gap cannot be reached until the PC structure is appropriate for generating both TE and TM gaps. The optimal combination of the frame construction and dielectric rods in this paper also agree with the rule of thumb that TE band gaps based on the dielectric rod and TM band gaps based on the dielectric hole [20

20. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B 61(20), 13458–13464 (2000). [CrossRef]

].

4. Conclusion

In summary, we report a new designed geometric structure of 2D PC with the maximum absolute band gap, as the knowledge about the author. The gradient information and GPM are employed to reduce the design variables obviously. Through numerical calculation, the best 2D PC structure is found with a large absolute band gap of 0.1623 (2πc/a) at a midfrequency of 0.8690 (2πc/a). Compared with the structures obtained by other methods, our structure is more close to the global optimal solution, because the GPM treatment adjusts the dielectric distribution by a continuous way rather than a discrete way. In order to keep the consistent with the practical manufacturing, we simplify the structure by the combination of triangle and rectangular geometry. After parameter optimization, we obtain a optimal 2D PC structure with absolute band gap of 0.1735 (2πc/a) at a midfrequency of 0.8828 (2πc/a). And the error analysis shows that the structure is stable for fabrication. The absolute band gap of this optimal structure is much larger than the best designs of the absolute band gap found in [2

2. L. F. Shen, S. He, and S. S. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B 66(16), 165315 (2002). [CrossRef]

5

5. L. F. Shen, Z. Ye, and S. L. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B 68(3), 035109 (2003). [CrossRef]

]. To the best of our knowledge, the optimal structure with absolute band gap of 0.1735 (2πc/a) in this paper has never been reported before.

Acknowledgments

Project supported by the National High Technology Research and Development Program of China (Grant No. 2009AA03Z405), the National Natural Science Foundation of China (Grant Nos. 60908028, 60971068, 10979065), the Fundamental Research Funds for the Central Universities (2009RC0411, 2009RC0412 and 2011RC0402), and Program for New Century Excellent Talents in University (NTCE-10-0261).

References and links

1.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]

2.

L. F. Shen, S. He, and S. S. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B 66(16), 165315 (2002). [CrossRef]

3.

H. P. Li, L. Y. Jiang, W. Jia, H. X. Qiang, and X. Y. Li, “Genetic optimization of two-dimensional photonic crystals for large absolute band-gap,” Opt. Commun. 282(14), 3012–3017 (2009). [CrossRef]

4.

S. Zarei, M. Shahabadi, and S. Mohajerzadeh, “Symmetry reduction for maximization of higher-order stop-bands in two-dimensional photonic crystals,” J. Mod. Opt. 55(18), 2971–2980 (2008). [CrossRef]

5.

L. F. Shen, Z. Ye, and S. L. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B 68(3), 035109 (2003). [CrossRef]

6.

M. Qiu and S. He, “Optimal design of a two-dimensional photonic crystal of square lattice with a large complete two-dimensional band gap,” J. Opt. Soc. Am. B 17(6), 1027–1030 (2000). [CrossRef]

7.

W. L. Liu and T. J. Yang, “Engineering the band-gap of a two-dimensional photonic crystal with slender dielectric veins,” Phys. Lett. A 369(5-6), 518–523 (2007). [CrossRef]

8.

F. Wen, S. David, X. Checoury, M. El Kurdi, and P. Boucaud, “Two-dimensional photonic crystals with large complete photonic band gaps in both TE and TM polarizations,” Opt. Express 16(16), 12278–12289 (2008). [CrossRef] [PubMed]

9.

O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett. 100(15), 153904 (2008). [CrossRef] [PubMed]

10.

H. Men, N. C. Nguyen, R. M. Freund, K. M. Lim, P. A. Parrilo, and J. Peraire, “Design of photonic crystals with multiple and combined band gaps,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(4), 046703 (2011). [CrossRef] [PubMed]

11.

J. Norato, R. Haber, D. Tortorelli, and M. P. Bendsoe, “A geometry projection method for shape optimization,” Int. J. Numer. Methods Eng. 60(14), 2289–2312 (2004). [CrossRef]

12.

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103(3), 033102 (2008). [CrossRef]

13.

S. Preble, M. Lipson, and H. Lipson, “Two-dimensional photonic crystals designed by evolutionary algorithms,” Appl. Phys. Lett. 86(6), 061111 (2005). [CrossRef]

14.

O. Sigmund and J. Petersson, “Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima,” Struct. Optim. 16(1), 68–75 (1998). [CrossRef]

15.

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett. 32(1), 77–79 (2007). [CrossRef] [PubMed]

16.

G. Turk and J. F. O’Brien, “Modeling with Implicit Surfaces that Interpolate,” ACM Trans. Graph. 21(4), 855–873 (2002). [CrossRef]

17.

H. Tian, Z. Yu, L. Han, and Y. Liu, “Birefringence and confinement loss properties in photonic crystal fibers under lateral stress,” IEEE Photon. Technol. Lett. 20(22), 1830–1832 (2008). [CrossRef]

18.

T. Hong-Da, Y. Zhong-Yuan, H. Li-Hong, and L. Yu-Min, “Lateral stress-induced propagation characteristics in photonic crystal fibres,” Chin. Phys. B 18(3), 1109–1115 (2009). [CrossRef]

19.

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett. 84(12), 2022–2024 (2004). [CrossRef]

20.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B 61(20), 13458–13464 (2000). [CrossRef]

OCIS Codes
(230.0230) Optical devices : Optical devices
(230.5298) Optical devices : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: August 15, 2011
Manuscript Accepted: September 1, 2011
Published: September 20, 2011

Citation
Donglin Wang, Zhongyuan Yu, Yumin Liu, Pengfei Lu, Lihong Han, Hao Feng, Xiaotao Guo, and Han Ye, "The optimal structure of two dimensional photonic crystals with the large absolute band gap," Opt. Express 19, 19346-19353 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19346


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References

  1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58(23), 2486–2489 (1987). [CrossRef] [PubMed]
  2. L. F. Shen, S. He, and S. S. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B66(16), 165315 (2002). [CrossRef]
  3. H. P. Li, L. Y. Jiang, W. Jia, H. X. Qiang, and X. Y. Li, “Genetic optimization of two-dimensional photonic crystals for large absolute band-gap,” Opt. Commun.282(14), 3012–3017 (2009). [CrossRef]
  4. S. Zarei, M. Shahabadi, and S. Mohajerzadeh, “Symmetry reduction for maximization of higher-order stop-bands in two-dimensional photonic crystals,” J. Mod. Opt.55(18), 2971–2980 (2008). [CrossRef]
  5. L. F. Shen, Z. Ye, and S. L. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B68(3), 035109 (2003). [CrossRef]
  6. M. Qiu and S. He, “Optimal design of a two-dimensional photonic crystal of square lattice with a large complete two-dimensional band gap,” J. Opt. Soc. Am. B17(6), 1027–1030 (2000). [CrossRef]
  7. W. L. Liu and T. J. Yang, “Engineering the band-gap of a two-dimensional photonic crystal with slender dielectric veins,” Phys. Lett. A369(5-6), 518–523 (2007). [CrossRef]
  8. F. Wen, S. David, X. Checoury, M. El Kurdi, and P. Boucaud, “Two-dimensional photonic crystals with large complete photonic band gaps in both TE and TM polarizations,” Opt. Express16(16), 12278–12289 (2008). [CrossRef] [PubMed]
  9. O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett.100(15), 153904 (2008). [CrossRef] [PubMed]
  10. H. Men, N. C. Nguyen, R. M. Freund, K. M. Lim, P. A. Parrilo, and J. Peraire, “Design of photonic crystals with multiple and combined band gaps,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(4), 046703 (2011). [CrossRef] [PubMed]
  11. J. Norato, R. Haber, D. Tortorelli, and M. P. Bendsoe, “A geometry projection method for shape optimization,” Int. J. Numer. Methods Eng.60(14), 2289–2312 (2004). [CrossRef]
  12. W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys.103(3), 033102 (2008). [CrossRef]
  13. S. Preble, M. Lipson, and H. Lipson, “Two-dimensional photonic crystals designed by evolutionary algorithms,” Appl. Phys. Lett.86(6), 061111 (2005). [CrossRef]
  14. O. Sigmund and J. Petersson, “Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima,” Struct. Optim.16(1), 68–75 (1998). [CrossRef]
  15. W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett.32(1), 77–79 (2007). [CrossRef] [PubMed]
  16. G. Turk and J. F. O’Brien, “Modeling with Implicit Surfaces that Interpolate,” ACM Trans. Graph.21(4), 855–873 (2002). [CrossRef]
  17. H. Tian, Z. Yu, L. Han, and Y. Liu, “Birefringence and confinement loss properties in photonic crystal fibers under lateral stress,” IEEE Photon. Technol. Lett.20(22), 1830–1832 (2008). [CrossRef]
  18. T. Hong-Da, Y. Zhong-Yuan, H. Li-Hong, and L. Yu-Min, “Lateral stress-induced propagation characteristics in photonic crystal fibres,” Chin. Phys. B18(3), 1109–1115 (2009). [CrossRef]
  19. J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett.84(12), 2022–2024 (2004). [CrossRef]
  20. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,” Phys. Rev. B61(20), 13458–13464 (2000). [CrossRef]

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