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Photonic band gaps of two-dimensional square-lattice photonic crystals based on 8-shaped scatters
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ARTICLE IN PRESS Optik xxx (2015) xxx–xxx
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Photonic band gaps of two-dimensional square-lattice photonic crystals based on 8-shaped scatters Kaibin Chu a , Quan Xu a,∗ , Kang Xie b , Cuiyun Peng a a b
School of Information Science & Engineering, Changzhou University, Jiangsu 213164, China School of Instrument Science and Opto-Electronics Engineering, HeFei University of Technology, Hefei 230009, China
a r t i c l e
i n f o
Article history: Received 14 April 2014 Accepted 25 May 2015 Available online xxx Keywords: Photonic crystals Photonic band gaps 8-Shaped Scatter
a b s t r a c t A novel photonic crystals (PhC) based on 8-shaped scatters with two-dimensional (2-D) square-lattice is presented. By employing the MIT photonic bands (MPB), the relationship between the structural parameters of the PhC and the properties of photonic band gaps (PBG) are investigated. The calculation results of 8-shaped scatters with 2-D square-lattice demonstrate that the reduction of scatters’ symmetry can produce larger number of PBG and broaden the width of PBG only for TM mode, and reversely for TE mode. The peculiarity would meet the distinguish employment of different mode. By optimizing the structural parameters, the maximum absolute PBG 0.1028(ωa/2c) at R = 0.3, εr = 11.56, = 45◦ is obtained and six absolute PBGs is got at R = 0.3, = 115◦ , εr ∈ [32.3, 33.5]. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Photonic crystals (PhC) are artificial structures with periodically structures and have photonic bandgaps (PBG) in which the propagation of photon is prohibited [1,2]. PhC with PBG have been widely studied as the basis of mirror [3], waveguides [4], lasers and cavities [5,6], filters [7,8], and other important components for high density optical integration. Based on the characteristics of the PBG, designing a PhC with larger PBG will produce significant impacts both in academic and application areas. Originally, most studies of PCs are based on square, triangular or rectangular lattice [9]. In order to more elastic design the PBG properties, the pursuer in the PhC literature struggle with PhC structures. Pervious research findings indicate that the low symmetrical lattice and low symmetrical scatter tend to provide the larger PBG. The PhC with parallelogram [10,11] and rhombus lattice [12–14] have been promoted to advance the width of PBG. Recently, the width of PBG has been improved by elliptic air holes [15], Taiji-shaped dielectric rods [16], dielectric rings [17], S-shaped rods [18] PhC. Inspired by the effects of shapes and orientations [19] of scatterers and lattice symmetries on the PBG in 2-D PhC, a novel PhC with 8-shaped scatters was proposed. This paper will present a summary of our ongoing research in the area of PBG based PhC applications. Therefore, the remainder of this paper is organized as follows: in
∗ Corresponding author. Tel.: +86 15151968306. E-mail address: [email protected] (Q. Xu).
Section 2 we present the structure of 8-shaped PhC and highlight the analytical method, in Section 3 the nature of PBG for different structural parameters are analyzed, in Section 4 we present the concluding remarks.
2. Description of the 8-shaped PhC and its analyze method Fig. 1 depicts the proposed structure of PhC implemented on a 2-D square-lattice consisting of 8-shaped scatters and its structural parameters. The lattice constant is a = 1 m, 8-shaped scatter with small circular radius r and the radius of the scatter R (with the unit of a), satisfy the relationship R = 2r. is the angle between the axis of the two small circular and x-axis, is positive while anticlockwise rolling. The 8-shaped scatters embedded in a air background with relative dielectric constant εr . Based on the theory of crystal = lattice + scatter, the first irreducible Brillouin zone (IBZ) must expand to the first Brillouin zone (BZ) because of the symmetry of scatter is lower than the symmetry of lattice. By employing the MIT photonic bands (MPB), the relationship between the structural parameters εr , R = 2r, of the PhC and the properties of photonic band gaps (PBG) are investigated. Fig. 2 shows the energy bands of the proposed PhC for TM mode achieved along the wave vector in the first BZ along the boundary. We suppose kx ∈ [− /a, /a] and divided into 20 intervals with sign the point from 1 to 21. In other words, 1 corresponding to −/a, 11 corresponding to 0 and 21 corresponding to −/a along kx axis. The expression of ky is the same as kx . Fig. 2a shows three dimensional scheme of the first band for TM mode (magnetic field
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Please cite this article in press as: K. Chu, et al., Photonic band gaps of two-dimensional square-lattice photonic crystals based on 8-shaped scatters, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.114
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Fig. 1. Schematic view of the proposed 2-D PhC structure with 8-shaped scatters.
parallel of the cylinders). The minimal value of the normalized frequency (NF) is zero when x = 11, y = 11 and maximal value of the NF is 0.29ωa/2c when x = 1, y = 1, x = 1, y = 21 and x = 21, y = 21. Fig. 2b shows the second band, the minimal value of the NF is 0.3438ωa/2c when x = 1, y = 11 and x = 21, y = 11, the maximal value of the NF is 0.4625ωa/2c when x = 11, y = 3 and x = 11, y = 19. With the same means, we get the minimal NF is 0.4586ωa/2c when x = 9, y = 1, x = 13, y = 1, x = 9, y = 21 and x = 13, y = 21, the maximal NF is 0.6706ωa/2c when x = 1, y = 11 and x = 21, y = 11 for the third band shown in Fig. 2c. The fourth band is shown in Fig. 2d, the minimal NF is 0.6419ωa/2c when x = 11, y = 5, and x = 11, y = 17, the maximal NF is 0.7068ωa/2c when x = 1, y = 10, x = 21, y = 10, x = 1, y = 12, and x = 21, y = 12. In conclusion, the maximal and minimal NF value of 8-shaped scatter PhC not appear at the edge of the first IBZ, but at the inner of the first BZ. The PBG between the first band and the second band is 0.0538ωa/2c. The PBG is studied in the first BZ consequently.
Fig. 2. The scheme diagrams of three dimensional bands for TM mode in the first BZ with εr = 11.56, R = 2r = 0.3, = 0◦ .
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Fig. 3. Schematic diagrams of structural parameters impact on absolute PBG. (a) and (b) keeping εr = 11.56, R = 0.3, ∈ [0◦ ,180◦ ]. (c) and (d) maintaining R = 0.3, = 115◦ , εr ∈ [1,40] (e) and (f) keeping εr = 11.56, = 115◦ , R ∈ [0.1,0.5]
3. Simulation and results The absolute PBG and it of TM mode and TE mode for different radius of the scatter R, anticlockwise rolling angle and relative dielectric constant εr are analyzed. The width and number of the PBG for 8-shaped scatters are given by reasonable change εr , R = 2r, . 3.1. Changed impact on width and numbers of absolute PBG Keeping εr = 11.56, R = 0.3 and changing ∈[0◦ , 180◦ ] with interval of 3◦ . The absolute PBG characteristics of 8-shaped scatters are shown in Fig. 3a for TM mode and Fig. 3b for TE mode. Investigation findings indicate that the TM mode exist maximum absolute PBG at = 45◦ and the width is 0.1028(ωa/2c). With structural parameters of εr = 11.56, R = 0.3, ∈ [132◦ , 138◦ ], five absolute PBGs are derived. In contrast, the number of absolute PBGs is less and the PBG is slim at ∈ [0◦ , 180◦ ] for TE mode. With R = 0.3, εr = 11.56, the width of maximum PBG for TM mode changing with is shown in Fig. 4a. The maximum PBG range is [0.0223ωa/2c, 0.1028ωa/2c]. In the range of ∈ [0◦ , 180◦ ], the maximum PBG has three minimal point. The maximum PBG are 0.0223ωa/2c with = 9◦ , 0.0538ωa/2c with = 9◦ and 0.0283ωa/2c with = 168◦ . With the scope of ∈ [0◦ , 180◦ ], the maximum PBG has two maximal point, there are 0.1028ωa/2c with = 45◦ and 0.0820ωa/2c with = 135◦ . Conclusion indicated that must approach the first and third quadrants or the second and fourth ones in order to obtain larger PBG for PhC with 8-shaped scatters.
Fig. 4. Maximum absolute PBG for TM mode with various structural parameters.
3.2. Changed relative dielectric constantεr impact on width and numbers of absolute PBG Maintaining R = 0.3, = 115◦ and changing εr ∈ [1,40] with interval of 0.06. The PWE is hired to obtain the absolute PBG characteristics of 8-shaped scatters. Fig. 3c shows the absolute PBG for TM mode and Fig. 3d for TE mode. Research shows that the maximum absolute PBG is 0.0611(ωa/2c) with εr = 29.9 for TM mode. There are six absolute PBGs with εr ∈ [32.3, 33.5]. The absolute PBGs move to low frequency as εr gets larger. Larger relative dielectric constant should be select to obtain larger PBG for 8-shaped scatters PhC. Obviously, the TE mode has single PBG in εr ∈ [1,40] and the absolute PBG move to lower frequency as εr gets larger. The characteristics of energy bands for TE are complicated simultaneously. With R = 0.3, = 115◦ , the width of maximum absolute PBG for TM mode changing with εr is shown in Fig. 4b. The range of the maximum absolute PBG is [0.0363ωa/2c, 0.0611ωa/2c].
Please cite this article in press as: K. Chu, et al., Photonic band gaps of two-dimensional square-lattice photonic crystals based on 8-shaped scatters, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.114
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3.3. Changed the radius of the scatter R effect on width and numbers of absolute PBG Keeping εr = 11.56, = 115◦ and changing R ∈ [0.1, 0.5] with interval of 0.01. The absolute PBG characteristics of 8-shaped scatters are shown in Fig. 3e for TM mode and Fig. 3f for TE mode. Investigation findings indicate that the TM mode exist maximum absolute PBG at R = 0.48 and the width is 0.0862(ωa/2c). The absolute PBG move to higher frequency as R gets larger. With the structural parameters of εr = 11.56, = 115◦ , R ∈ [0.36, 0.39], four absolute PBGs are derived. In contrast, the number of absolute PBGs is less and the PBG is slim at R ∈ [0.1, 0.5] for TE mode. With εr = 11.56, = 115◦ , the width of maximum absolute PBG for TM mode changing with R is shown in Fig. 4c. The range of the maximum absolute PBG is [0.0375ωa/2c, 0.0862ωa/2c]. All the analysis shows that 8-shaped scatters with 2-D squarelattice can produce larger width and number of PBG only for TM mode, but reversely for TE mode. The research results show that the energy bands are complicated for TE mode, and purpose to establish solid foundations for the application based on energy band characteristics. 4. Conclusions The absolute PBG of 8-shaped scatters with different structural parameters (the radius of the scatter R, the angle between the axis of the two small circular and x-axis and relative dielectric constant εr ) has been presented. Research findings indicate that absolute PBG move to higher frequency as εr or R gets larger and exist symmetrical characteristics with changed. With given structural parameters, the maximum absolute PBG 0.1028(ωa/2c) at R = 0.3, εr = 11.56, = 45◦ is obtained and the maximum number of six absolute PBGs R = 0.3, = 115◦ , εr ∈ [32.3, 33.5]. By optimizing design the structural parameters, more widen maximum absolute PBG would be obtained. Although the results are based on arbitrary structural parameters, they are sufficient to show the conceptual framework of the proposed 8-shaped scatters PhC and establish solid foundations for the photonic crystal apparatus design based on 8-shaped scatters two-dimensional square-lattice PhC. Acknowledgments The author would like to acknowledge Project supported by the Natural Science Foundations for colleges and universities in Jiangsu
Province through Grant no. 14KJB430004, Changzhou University Science Foundation for Youths through Grant no. ZMF1102075 and Project supported by the National Natural Science Foundation of China (Grant no. 60588502). References [1] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059–2062. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58 (1987) 2486–2489. [3] A. Bruyant, G. Lérondel, P.J. Reece, M. Gal, Allsilicon omnidirectional mirrors based on one-dimensional photonic crystals, Appl. Phys. Lett. 82 (2003) 3227–3229. [4] A. Chutinan, S. Noda, Waveguides and waveguide bends in two-dimensional photonic crystal slabs, Phys. Rev. B 62 (2000) 4488–4492. [5] Y. Akahane, T. Asano, B.S. Song, High-Q photonic nanocavity in a twodimensional photonic crystal, Nat. (Lond.) 425 (2003) 944–947. [6] T.D. Happ, A. Markard, M. Kamp, et al., InP-based short cavity lasers with 2D photonic crystal mirror, Electron. Lett. 37 (7) (2001) 428–429. [7] Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, S. Noda, Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab, Appl. Phys. Lett. 82 (2003) 1341–1343. [8] M. Qiu, B. Jaskorzynska, Design of a channel drop filter in a twodimensional triangular photonic crystal, Appl. Phys. Lett. 83 (2003) 1074–1076. [9] J. Mizuguchi, Y. Tanaka, S. Tamura, M. Notomi, Focusing of light in a threedimensional cubic photonic crystal, Phys. Rev. B 67 (2003) 075106–075109. [10] Y. Ogawa, Y. Omura, Study on self-collimated light focusing device using the 2-D photonic crystal with a parallelogram lattice, J. Lightwave Technol. 23 (12) (2005) 4374–4381. [11] D. Gao, Z. Zhou, D.S. Citrin, Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice, J. Opt. Soc. Am. A 25 (3) (2008) 791–795. [12] Y.C. Chuang, T.J. Suleski, Complex rhombus lattice photonic crystals for broadband all-angle self-collimation, J. Opt. 12 (3) (2010) 035102–035107. [13] L. Qi, Z. Yang, X. Gao, et al., Analysis of two-dimensional photonic band gap structure with a rhombus lattice, Chin. Opt. Lett. 6 (4) (2008) 279–281. [14] Q. Xu, C.Y. Peng, The properties of two dimensional photonic crystals bandgap structure with rhombus lattice, Optik 125 (1) (2013) 104–106. [15] M. Qiu, S.L. He, Large complete band gap in two-dimensional photonic crystals with elliptic air holes, Phys. Rev. B 60 (15) (1999) 10610–10612. [16] Juan-Juan Sheng, Xing-Dao He, Bin Liu, et al., Photonic band gaps of twodimensional hexagon-lattice photonic crystals based no Taiji-shaped dielectric rods, Acta Phys. Sin. 62 (8) (2013) 4213–4217. [17] Q. Xu, K. Xie, H.J. Yang, Self collimation in square lattice two dimensional photonic crystals with ring-shaped holes, IEEE Int. Conf. Nano Sci. Technol. 5 (2010) 182–186. [18] A.F. Oskooi, D. Roundy, M. Ibanescu, et al., Meep: a flexible free-software package for electromagnetic simulations by the FDTD method, Comput. Phys. Commun. 181 (3) (2010) 687–702. [19] R. Wang, X. Wang, B. Gu, et al., Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals, J. Appl. Phys. 90 (9) (2001) 4307–4313.
Please cite this article in press as: K. Chu, et al., Photonic band gaps of two-dimensional square-lattice photonic crystals based on 8-shaped scatters, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.114
ARTICLE IN PRESS Optik xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Photonic band gaps of two-dimensional square-lattice photonic crystals based on 8-shaped scatters Kaibin Chu a , Quan Xu a,∗ , Kang Xie b , Cuiyun Peng a a b
School of Information Science & Engineering, Changzhou University, Jiangsu 213164, China School of Instrument Science and Opto-Electronics Engineering, HeFei University of Technology, Hefei 230009, China
a r t i c l e
i n f o
Article history: Received 14 April 2014 Accepted 25 May 2015 Available online xxx Keywords: Photonic crystals Photonic band gaps 8-Shaped Scatter
a b s t r a c t A novel photonic crystals (PhC) based on 8-shaped scatters with two-dimensional (2-D) square-lattice is presented. By employing the MIT photonic bands (MPB), the relationship between the structural parameters of the PhC and the properties of photonic band gaps (PBG) are investigated. The calculation results of 8-shaped scatters with 2-D square-lattice demonstrate that the reduction of scatters’ symmetry can produce larger number of PBG and broaden the width of PBG only for TM mode, and reversely for TE mode. The peculiarity would meet the distinguish employment of different mode. By optimizing the structural parameters, the maximum absolute PBG 0.1028(ωa/2c) at R = 0.3, εr = 11.56, = 45◦ is obtained and six absolute PBGs is got at R = 0.3, = 115◦ , εr ∈ [32.3, 33.5]. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Photonic crystals (PhC) are artificial structures with periodically structures and have photonic bandgaps (PBG) in which the propagation of photon is prohibited [1,2]. PhC with PBG have been widely studied as the basis of mirror [3], waveguides [4], lasers and cavities [5,6], filters [7,8], and other important components for high density optical integration. Based on the characteristics of the PBG, designing a PhC with larger PBG will produce significant impacts both in academic and application areas. Originally, most studies of PCs are based on square, triangular or rectangular lattice [9]. In order to more elastic design the PBG properties, the pursuer in the PhC literature struggle with PhC structures. Pervious research findings indicate that the low symmetrical lattice and low symmetrical scatter tend to provide the larger PBG. The PhC with parallelogram [10,11] and rhombus lattice [12–14] have been promoted to advance the width of PBG. Recently, the width of PBG has been improved by elliptic air holes [15], Taiji-shaped dielectric rods [16], dielectric rings [17], S-shaped rods [18] PhC. Inspired by the effects of shapes and orientations [19] of scatterers and lattice symmetries on the PBG in 2-D PhC, a novel PhC with 8-shaped scatters was proposed. This paper will present a summary of our ongoing research in the area of PBG based PhC applications. Therefore, the remainder of this paper is organized as follows: in
∗ Corresponding author. Tel.: +86 15151968306. E-mail address: [email protected] (Q. Xu).
Section 2 we present the structure of 8-shaped PhC and highlight the analytical method, in Section 3 the nature of PBG for different structural parameters are analyzed, in Section 4 we present the concluding remarks.
2. Description of the 8-shaped PhC and its analyze method Fig. 1 depicts the proposed structure of PhC implemented on a 2-D square-lattice consisting of 8-shaped scatters and its structural parameters. The lattice constant is a = 1 m, 8-shaped scatter with small circular radius r and the radius of the scatter R (with the unit of a), satisfy the relationship R = 2r. is the angle between the axis of the two small circular and x-axis, is positive while anticlockwise rolling. The 8-shaped scatters embedded in a air background with relative dielectric constant εr . Based on the theory of crystal = lattice + scatter, the first irreducible Brillouin zone (IBZ) must expand to the first Brillouin zone (BZ) because of the symmetry of scatter is lower than the symmetry of lattice. By employing the MIT photonic bands (MPB), the relationship between the structural parameters εr , R = 2r, of the PhC and the properties of photonic band gaps (PBG) are investigated. Fig. 2 shows the energy bands of the proposed PhC for TM mode achieved along the wave vector in the first BZ along the boundary. We suppose kx ∈ [− /a, /a] and divided into 20 intervals with sign the point from 1 to 21. In other words, 1 corresponding to −/a, 11 corresponding to 0 and 21 corresponding to −/a along kx axis. The expression of ky is the same as kx . Fig. 2a shows three dimensional scheme of the first band for TM mode (magnetic field
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Please cite this article in press as: K. Chu, et al., Photonic band gaps of two-dimensional square-lattice photonic crystals based on 8-shaped scatters, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.114
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Fig. 1. Schematic view of the proposed 2-D PhC structure with 8-shaped scatters.
parallel of the cylinders). The minimal value of the normalized frequency (NF) is zero when x = 11, y = 11 and maximal value of the NF is 0.29ωa/2c when x = 1, y = 1, x = 1, y = 21 and x = 21, y = 21. Fig. 2b shows the second band, the minimal value of the NF is 0.3438ωa/2c when x = 1, y = 11 and x = 21, y = 11, the maximal value of the NF is 0.4625ωa/2c when x = 11, y = 3 and x = 11, y = 19. With the same means, we get the minimal NF is 0.4586ωa/2c when x = 9, y = 1, x = 13, y = 1, x = 9, y = 21 and x = 13, y = 21, the maximal NF is 0.6706ωa/2c when x = 1, y = 11 and x = 21, y = 11 for the third band shown in Fig. 2c. The fourth band is shown in Fig. 2d, the minimal NF is 0.6419ωa/2c when x = 11, y = 5, and x = 11, y = 17, the maximal NF is 0.7068ωa/2c when x = 1, y = 10, x = 21, y = 10, x = 1, y = 12, and x = 21, y = 12. In conclusion, the maximal and minimal NF value of 8-shaped scatter PhC not appear at the edge of the first IBZ, but at the inner of the first BZ. The PBG between the first band and the second band is 0.0538ωa/2c. The PBG is studied in the first BZ consequently.
Fig. 2. The scheme diagrams of three dimensional bands for TM mode in the first BZ with εr = 11.56, R = 2r = 0.3, = 0◦ .
Please cite this article in press as: K. Chu, et al., Photonic band gaps of two-dimensional square-lattice photonic crystals based on 8-shaped scatters, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.114
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Fig. 3. Schematic diagrams of structural parameters impact on absolute PBG. (a) and (b) keeping εr = 11.56, R = 0.3, ∈ [0◦ ,180◦ ]. (c) and (d) maintaining R = 0.3, = 115◦ , εr ∈ [1,40] (e) and (f) keeping εr = 11.56, = 115◦ , R ∈ [0.1,0.5]
3. Simulation and results The absolute PBG and it of TM mode and TE mode for different radius of the scatter R, anticlockwise rolling angle and relative dielectric constant εr are analyzed. The width and number of the PBG for 8-shaped scatters are given by reasonable change εr , R = 2r, . 3.1. Changed impact on width and numbers of absolute PBG Keeping εr = 11.56, R = 0.3 and changing ∈[0◦ , 180◦ ] with interval of 3◦ . The absolute PBG characteristics of 8-shaped scatters are shown in Fig. 3a for TM mode and Fig. 3b for TE mode. Investigation findings indicate that the TM mode exist maximum absolute PBG at = 45◦ and the width is 0.1028(ωa/2c). With structural parameters of εr = 11.56, R = 0.3, ∈ [132◦ , 138◦ ], five absolute PBGs are derived. In contrast, the number of absolute PBGs is less and the PBG is slim at ∈ [0◦ , 180◦ ] for TE mode. With R = 0.3, εr = 11.56, the width of maximum PBG for TM mode changing with is shown in Fig. 4a. The maximum PBG range is [0.0223ωa/2c, 0.1028ωa/2c]. In the range of ∈ [0◦ , 180◦ ], the maximum PBG has three minimal point. The maximum PBG are 0.0223ωa/2c with = 9◦ , 0.0538ωa/2c with = 9◦ and 0.0283ωa/2c with = 168◦ . With the scope of ∈ [0◦ , 180◦ ], the maximum PBG has two maximal point, there are 0.1028ωa/2c with = 45◦ and 0.0820ωa/2c with = 135◦ . Conclusion indicated that must approach the first and third quadrants or the second and fourth ones in order to obtain larger PBG for PhC with 8-shaped scatters.
Fig. 4. Maximum absolute PBG for TM mode with various structural parameters.
3.2. Changed relative dielectric constantεr impact on width and numbers of absolute PBG Maintaining R = 0.3, = 115◦ and changing εr ∈ [1,40] with interval of 0.06. The PWE is hired to obtain the absolute PBG characteristics of 8-shaped scatters. Fig. 3c shows the absolute PBG for TM mode and Fig. 3d for TE mode. Research shows that the maximum absolute PBG is 0.0611(ωa/2c) with εr = 29.9 for TM mode. There are six absolute PBGs with εr ∈ [32.3, 33.5]. The absolute PBGs move to low frequency as εr gets larger. Larger relative dielectric constant should be select to obtain larger PBG for 8-shaped scatters PhC. Obviously, the TE mode has single PBG in εr ∈ [1,40] and the absolute PBG move to lower frequency as εr gets larger. The characteristics of energy bands for TE are complicated simultaneously. With R = 0.3, = 115◦ , the width of maximum absolute PBG for TM mode changing with εr is shown in Fig. 4b. The range of the maximum absolute PBG is [0.0363ωa/2c, 0.0611ωa/2c].
Please cite this article in press as: K. Chu, et al., Photonic band gaps of two-dimensional square-lattice photonic crystals based on 8-shaped scatters, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.114
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ARTICLE IN PRESS K. Chu et al. / Optik xxx (2015) xxx–xxx
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3.3. Changed the radius of the scatter R effect on width and numbers of absolute PBG Keeping εr = 11.56, = 115◦ and changing R ∈ [0.1, 0.5] with interval of 0.01. The absolute PBG characteristics of 8-shaped scatters are shown in Fig. 3e for TM mode and Fig. 3f for TE mode. Investigation findings indicate that the TM mode exist maximum absolute PBG at R = 0.48 and the width is 0.0862(ωa/2c). The absolute PBG move to higher frequency as R gets larger. With the structural parameters of εr = 11.56, = 115◦ , R ∈ [0.36, 0.39], four absolute PBGs are derived. In contrast, the number of absolute PBGs is less and the PBG is slim at R ∈ [0.1, 0.5] for TE mode. With εr = 11.56, = 115◦ , the width of maximum absolute PBG for TM mode changing with R is shown in Fig. 4c. The range of the maximum absolute PBG is [0.0375ωa/2c, 0.0862ωa/2c]. All the analysis shows that 8-shaped scatters with 2-D squarelattice can produce larger width and number of PBG only for TM mode, but reversely for TE mode. The research results show that the energy bands are complicated for TE mode, and purpose to establish solid foundations for the application based on energy band characteristics. 4. Conclusions The absolute PBG of 8-shaped scatters with different structural parameters (the radius of the scatter R, the angle between the axis of the two small circular and x-axis and relative dielectric constant εr ) has been presented. Research findings indicate that absolute PBG move to higher frequency as εr or R gets larger and exist symmetrical characteristics with changed. With given structural parameters, the maximum absolute PBG 0.1028(ωa/2c) at R = 0.3, εr = 11.56, = 45◦ is obtained and the maximum number of six absolute PBGs R = 0.3, = 115◦ , εr ∈ [32.3, 33.5]. By optimizing design the structural parameters, more widen maximum absolute PBG would be obtained. Although the results are based on arbitrary structural parameters, they are sufficient to show the conceptual framework of the proposed 8-shaped scatters PhC and establish solid foundations for the photonic crystal apparatus design based on 8-shaped scatters two-dimensional square-lattice PhC. Acknowledgments The author would like to acknowledge Project supported by the Natural Science Foundations for colleges and universities in Jiangsu
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Please cite this article in press as: K. Chu, et al., Photonic band gaps of two-dimensional square-lattice photonic crystals based on 8-shaped scatters, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.114