- Email: [email protected]
Effects of noncircular inner scatterers on absolute photonic band gap of 2D triangular annular photonic crystals
Accepted Manuscript Title: Effects of noncircular inner scatterers on absolute photonic band gap of 2D triangular annular photonic crystals Authors: Dan Liu, Sen Hu PII: DOI: Reference:
S0030-4026(17)30181-X http://dx.doi.org/doi:10.1016/j.ijleo.2017.02.038 IJLEO 58854
To appear in: Received date: Accepted date:
24-11-2016 10-2-2017
Please cite this article as: Dan Liu, Sen Hu, Effects of noncircular inner scatterers on absolute photonic band gap of 2D triangular annular photonic crystals, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2017.02.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effects of noncircular inner scatterers on absolute photonic band gap of 2D triangular annular photonic crystals Dan Liu*, Sen Hu Department of Physics and Mechanical & Electrical Engineering, Hubei University of Education, Wuhan, 430205, China
*Corresponding author. E-mail: [email protected] .
Abstract Using the plane wave expansion method, the photonic band gap (PBG) structures of quasi 2D triangular annular photonic crystals (PCs) with square and hexagonal inner scatterers have been calculated. We investigate the effects of structural parameters on absolute PBGs for silicon (Si)-based and germanium (Ge)-based annular PCs. The results reveal that the optimal inner scatterer shape is hexagonal. Moreover, we discuss the PBGs as a function of inner scatterer orientation. As for silicon annular PCs, utilization of high refractive index or anisotropic tellurium (Te) rods can lead to the formation of larger absolute PBGs. Keywords: Absolute photonic band gap; Silicon annular photonic crystal; Anisotropic tellurium 1. Introduction Within the past decades, there has been great interest in the study of PCs [1,2], both theoretically and experimentally, due to their unique ability to control the flow of light at optical wavelength scale [3]. PCs, periodic dielectric materials with a photonic band gap, enable one to control the optical emission properties of materials placed inside them. Many of the promising applications of two- and three- dimensional PCs depend on the location and width of PBGs [4-9]. Although three-dimensional (3D) PCs suggest the most interesting idea for novel applications, two-dimensional (2D) PCs have also been strongly studied, since they can be fabricated more easily than 3D ones and may be employed in optical and electronic devices [10]. Moreover, 2D structures were widely utilized in light emitting 1
diodes, low-threshold lasers, all-optical switches, and high-Q microcavities [11-15]. Therefore, significant attention has been drawn to design and fabrication of 2D PCs. The modes in 2D PCs can be classified as transverse-electric (TE) or transverse-magnetic (TM). An absolute PBG exists for PC only when PBGs are present in both polarizations and overlap with each other. Obtaining a large absolute PBG is very important for various applications [14]. Therefore, it is significant to study the characteristics of the PBG and design the structure with PBG as wide as possible. To our knowledge, employing an anisotropic material or magnetic material, structural deformation, composite structures and changing the shape of scatterer are all the approaches to obtain wide PBG [16-19]. Recently, it has been reported that the suggested class of the so-called annular PCs, where dielectric rods are embedded into air holes of larger radius, have show large PBGs [11]. The designed structure is relatively simple and effective to control PC properties for different polarization. Then, there are many studies on the annular PCs. Most of the works analyze the structure with circle rods embedded into circle air holes. However, if we introduce the noncircular scatterers into the annular PCs, will change the photonic band structure, and may obtain a large PBG. Moreover, silicon PCs are now attracting much attention. A class of integrated optical devices is built in the platform of the structure [20-22]. Especially, the creation of large absolute PBGs in certain frequency regions is an important issue for the design of most optical devices. Therefore, we pay more attention to the 2D silicon annular PCs, and hope to find the effective methods to obtain largest absolute PBGs. In this paper, we investigate quasi 2D triangular annular PCs with square and hexagonal inner scatterers. First, using the plane wave expansion (PWE) method, we compute the absolute PBGs for Si-based and Ge-based annular PCs. The results reveal that silicon annular PCs, regardless of circular, square, or hexagonal inner rods, are unable to possess a larger absolute gap than triangular air-hole PCs, and have small gap size. However, Ge-based annular PCs will present larger absolute gaps than the air-hole PCs in certain conditions. Moreover, the results indicate that the annular 2
PC with hexagonal inner scatterers is the optimal structure. Secondly, the effect of inner scatterer orientation on the PBGs is investigated. The result shows the optimal angle value. Finally, we find that the gap significantly increase by using high refractive index or Te rods instead of Si rods in silicon annular PCs. 2. Theory and model In order to determine absolute PBGs in periodic photonic crystal structures, we start with the wave equation for the magnetic field. In inhomogeneous dielectric materials, Maxwell’ equations for the magnetic field can be written in frequency domain as [14,23] 1 2 H(r) 2 H(r) (r) c
(1)
where is the frequency and c is the speed of light. Because of the relation H(r) 0 , H( r ) is transverse. For periodic systems, the magnetic field can be
expressed as a sum of plane waves [24]: 2
H(r) hGei (k+G)r hG, e ei (k+G)r G
(2)
G 1
where k is a wave vector in the first Brillouin zone, G is a reciprocal lattice vector and e ( =1,2) are unit vectors perpendicular to k G . So Eq. (1) is expressed as a linear matrix equation for the dispersion of EM waves [24-26]:
hG,1 2 hG,1 H 2 G,G G hG,2 c hG,2
(3)
where
e e e 2 e 1 H G,G k+G k+G 1 (G-G) 2 2 e1 e 2 e 1e 1
(4)
1( G) is the Fourier transform of the inverse of (r ) . The quantities 1( G) play an important role in the determination of the photonic band structure for both polarizations. We can write
1 (G)
1 1 (r )e-iGr d 2r cell 3
(5)
where is the area of the unit cell. In a 2D photonic crystal， k+G is in the xy plane for all G ’s and we can choose e1 and e 2 so that e1 is parallel to the z axis and e 2 is in the xy plane. In this case, e2 e1 = e1 e2 =0 and the matrix equation (4) splits into two uncoupled scalar equations. This gives rise to two polarizations of the Bloch waves. In the case of E-polarization ( E(r) is parallel to the z axis), hG,1 0 for all G and we have
k+G k+G 1 (G-G)hG, 2 G
2 c2
hG, 2
(6)
For the H-polarization ( H(r) is in the z direction), hG, 2 0 for all G and we have
(k+G) (k+G) G
1
(G-G)hG,1
2 c2
hG,1
(7)
The photonic band structure can be obtained by solving Eqs. (6) and (7). In this paper, we have considered two kinds of quasi triangular annular PCs with square and hexagonal inner scatterers, respectively. The corresponding models are shown in Fig. 1(a) and (b). The half-side length of square and the side length of hexagon are both expressed as r . The scatterers with the refractive index n are inserted in the middle of the air holes with radius R . The refractive index of the dielectric background is n , and a is the lattice constant. The conventional annular PCs with circle inner rods (radius r ) are also shown in Fig. 1(c). The orientation of noncircular rods relative to the lattice axis is defined by the angle , as shown in Fig. 1(d). In 2D PCs, we always decompose the electromagnetic modes into two types. TE modes have H normal to the plane and E in the plane. TM modes have just the reverse: E normal to the plane and H in the plane. The band structures for TE and TM modes can be completely different. It is possible that there are photonic band gaps for one polarization but not for the other [5]. An absolute PBG exists for a 2D photonic crystal only when PBGs in both polarization modes are present and overlap [14].
4
3. Results and discussion In this study, we consider quasi 2D triangular annular PCs, as shown in Fig. 1. First, we analyze the case of the material: Si-Si. The first material designates the background material, and the second one is for the rods. Our purpose here is to study the modification of the absolute PBG by the three geometrical parameters ( R , r , and
). Furthermore, we want to compare the absolute PBG width of three annular PCs with circular, square and hexagonal inner rods. This process is repeated for another case of Ge-Ge, considering the effects of the refractive index on the PBGs. First, we focus on the silicon annular PCs, and that is the case of Si-Si. The refractive index of Si is 3.45. Three annular PCs with circular, square and hexagonal dielectric rods are involved. The absolute PBGs as a function of the inner rod size r for several discrete values of the air-hole radius R are presented in Fig. 2. The relative width of the band gap is expressed as the gap-midgap ratio / m , where
m is the central frequency of the band gap and is the frequency width. The curves of three different models show that, for a fixed value of R and when the rod size r increases, the absolute PBG size first decreases and then oscillates attaining one maximum that is smaller than the starting PBG value. We know that, when r 0 , the model is, in fact, the usual air-hole PCs with a triangular lattice. The result reveals that the Si-Si structure, regardless of circular, square, or hexagonal inner rods, are unable to possess a larger absolute gap than triangular air-hole PCs. Moreover, it can be seen from Fig. 2 that annular PCs with circular rods show slightly wider PBGs than those with square rods. However, in contrast, annular PCs with hexagonal rods would be the optimal silicon annular PC structure to obtain large PBGs. Next, we investigate the PBG of a triangular annular PC with the case of Ge-Ge. The refractive index is 4.0 for Ge. Similarly to the case of Si-Si, here we study the effect of rod shape on the absolute PBG. The results are shown in Fig. 3. It indicates that, with the fixed R and increasing r , the absolute PBG size of three annular PCs first decreases and then oscillates attaining another maximum. However, the obvious 5
difference to Fig. 2 is that in this case the second maximum significantly increases and will higher than the starting PBG value in certain conditions. That is to say, for a certain value of R , Ge-based annular PCs will possess larger absolute gaps than the air-hole PCs. Additionally, we know that if r is too small, PCs will be harder to fabricate. Therefore, we should pay more attention to the second maximum with larger value of r . It can be seen from Fig. 3 that the annular PC with hexagonal inner scatterers is the optimal structure because it presents obviously wider PBGs than the other PC structures when r is larger than 0.1a. And the largest absolute gap of 21.84% is obtained with R = 0.47a and r =0.15a. Now, the effect of rod rotation ( ) on PBG is investigated for quasi triangular Ge-based annular PCs. First, we discuss gap width versus rotation of square inner rods. The orientation of square rods relative to the lattice axis is shown in Fig. 1(d). We fixed R =0.47a and r =0.12a. Square meets the four rotational symmetry operation, and rotation period is 90。. The rotation angle increase from 0。 to 45。, the rotating step size is 5。, and the result is shown in Fig. 4(a). The figure shows that the largest PBG presents at 0。. The absolute PBG size will slowly decrease with the increasing . And then we analyze gap width versus rotation of hexagonal inner rods. We fixed R =0.47a, r =0.15a, and set the orientation angle of Fig. 1(b) to 0。. Hexagon meets the six rotational symmetry operation, and rotation period is 60。. Fig. 4(b) shows that the largest PBG presents at 0。. When increases in a period, the absolute PBG size first decreases and then increases. The gap size will be smallest at the half period ( 30。). In summary, the triangular annular PC as shown in Fig. 1(b) is the optimal structure of the involved models. However, for Si-based annular PCs, in order to obtain larger absolute PBGs, we should take other special measures. Here, we keep the silicon background ( n =3.45), and involve hexagonal inner rods with different refractive index n . We analyze absolute PBGs variation with inner rod size, when
6
n increases from 2 to 6 with the increment of 1. The air-hole radius value R is 0.45a .
Shown in Fig. 5, when n is less than 4, the absolute gap will decrease to zero with the increasing r . However, if the refractive index increases from 4 to 6, the absolute gap width first decreases and then oscillates attaining another maximum, which is obviously higher than the starting PBG value. Moreover, compared with silicon annular PCs of Fig. 2, we indicate that introducing high refractive index inner rods in silicon background can effectively increase the absolute gap. When n =6, absolute gap reaches to 16.37% as r = 0.08 a . Furthermore, we involve the anisotropic material Te in the PCs, consider the case of Si-Te. The anisotropic Te has two different principal refractive indexes: the ordinary refractive index no 4.8 and the extraordinary refractive index ne 6.2 . Fig. 6 illustrates absolute PBG as a function of r . It can be seen from Fig. 6, the Si-Te structure present larger PBGs than those of the conventional air-hole PCs. Compared with Fig. 2(c), we find that the gap significantly increase by involving the Te rods. The largest absolute gap of 18.43% is obtained when R =0.45a and
r =0.07a. As a result, involving high refractive index or anisotropic inner rods in silicon background, is an effective method to obtain largest absolute PBGs for silicon PCs. 4. Conclusions In conclusion, we analyze 2D triangular annular PCs with square, hexagonal and circular inner scatterers. The effects of structural parameters on absolute PBGs have been investigated. The study indicates that the optimal inner scatterer is hexagonal. Considering the fabrication of PCs, we present the optimum inner rod size, air-hole radius, and rotation angle. Specially, for silicon annular PCs, absolute PBGs significantly increase by using high refractive index or anisotropic Te rods instead of Si rods. The creation of large PBGs in certain frequency regions should be helpful in designing most optical devices. Acknowledgements
7
This work was supported by the National Natural Science Foundation of China (Grant No. 11504100), and the fund for excellent youths of Hubei Provincial Department of Education (Grant No. Q20153004). References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486. [3] M. Turduev, I. H. Giden, H. Kurt, J. Opt. Soc. Am. B 29 (2012) 1589. [4] H. Kurt, D. Yilmaz, A.E. Akosman, Opt. Express 20 (2012) 20635. [5] J.D. Joannopoulos, R.D. Mead, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, 1995. [6] H. Wu, D.S. Citrin, L.Y. Jiang, Appl. Phys. Lett. 102 (2013) 141112. [7] J.D. Cox, M.R. Singh, G. Gumbs, M.A. Anton, F. Carreno, Phys. Rev. B 86 (2012) 125452. [8] L. Feng, M. Ayache, J. Huang, Science 333 (2011) 729. [9] A. Hatef, M.R. Singh, Phys. Rev. A 81 (2010) 063816. [10] T. F. Khalkhali, B. Rezaei, M. Kalafi, Opt. Commun. 284 (2011) 3315. [11] H. Kurt, D. S. Citrin, Opt. Express 13 (2005) 10316. [12] D. Liu, Y. H. Gao, D. S. Gao, X. Y. Han, Opt. Commun. 285 (2012) 1988. [13] S. J. An, J. H. Chae, G. -C. Yi, G. H. Park, Appl. Phys. Lett. 92 (2008) 121108. [14] D. Liu, Y. H. Gao, Aihong Tong, Sen Hu, Phys. Lett. A 379 (2015) 214. [15] Y. H. Park, Y. H. Shin, S. J. Noh, Y. M. Kim, S. S. Lee, C. G. Kim, K.S. An, C.Y. Park, Appl. Phys. Lett. 91 (2007) 012102. [16] Z. Y. Li, L. L. Lin, B. Y. Gu, G. Z. Yang, Physica B 279 (2000) 159. [17] B. Rezaei, T. F. Khalkhali, A. S. Vala, M. Kalafi, Opt. Commun. 282 (2009) 2861. [18] J. Hou, D. S. Citrin, H. Wu, Opt. Lett. 36 (2011) 2263. [19] T. F. Khalkhali, B. Rezaei, A. S. Vala, M. Kalafi, Appl. Opt. 52 (2013) 3745. [20] L. Gan, C.Z. Zhou, C. Wang, R.-J. Liu, D.-Z. Zhang, Z.-Y. Li, Phys. Status Solidi A 207 (2010) 2715. [21] Y.L. Fu, X.Y. Hu, Q.H. Gong, Phys. Lett. A 377 (2013) 329. 8
[22] J.F. Tao, J. Wu, H. Cai, Q.X. Zhang, J.M. Tsai, J.T. Lin, A.Q. Liu, Appl. Phys. Lett. 100 (2012) 113104. [23] D. Liu, H. Liu, Y. H. Gao, Solid State Commun. 172 (2013) 10. [24] T. Pan, F. Zhuang, Z.Y. Li, Solid State Commun. 129 (2004) 501. [25] K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65 (1990) 3152. [26] M. Plihal, A.A. Maradudin, Phys. Rev. B 44 (1991) 8565.
9
Figures Caption Fig.1. The 2D triangular annular PCs with different inner scatterer shapes: (a) Square; (b) Hexagonal; (c) Circular. (d) Rotated inner rods with angle . Fig.2. The absolute PBG variation as the inner rod size for Si-based annular PCs with different inner scatterer shapes: (a) Circular; (b) Square; (c) Hexagonal. Fig.3. The absolute PBG variation as the inner rod size for Ge-based annular PCs with different inner scatterer shapes: (a) Circular; (b) Square; (c) Hexagonal. Fig.4. Gap width versus (a) rotation angle of square inner rods for R =0.47a and r =0.12a, and (b) rotation angle of hexagonal inner rods for R =0.47a and r =0.15a in Ge-based annular PCs. Fig.5. The absolute PBG variation as the hexagonal inner rod size for R = 0.45a and n =3.45. Fig.6. The absolute PBG variation as the hexagonal inner rod size for Si-Te structures.
10
(a)
(b)
(c)
Fig. 1
11
(d)
(a)
(b)
(c) Fig. 2
12
(a)
(b)
(c) Fig. 3
13
(a)
(b) Fig. 4
14
Fig. 5
15
Fig. 6
16
S0030-4026(17)30181-X http://dx.doi.org/doi:10.1016/j.ijleo.2017.02.038 IJLEO 58854
To appear in: Received date: Accepted date:
24-11-2016 10-2-2017
Please cite this article as: Dan Liu, Sen Hu, Effects of noncircular inner scatterers on absolute photonic band gap of 2D triangular annular photonic crystals, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2017.02.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effects of noncircular inner scatterers on absolute photonic band gap of 2D triangular annular photonic crystals Dan Liu*, Sen Hu Department of Physics and Mechanical & Electrical Engineering, Hubei University of Education, Wuhan, 430205, China
*Corresponding author. E-mail: [email protected] .
Abstract Using the plane wave expansion method, the photonic band gap (PBG) structures of quasi 2D triangular annular photonic crystals (PCs) with square and hexagonal inner scatterers have been calculated. We investigate the effects of structural parameters on absolute PBGs for silicon (Si)-based and germanium (Ge)-based annular PCs. The results reveal that the optimal inner scatterer shape is hexagonal. Moreover, we discuss the PBGs as a function of inner scatterer orientation. As for silicon annular PCs, utilization of high refractive index or anisotropic tellurium (Te) rods can lead to the formation of larger absolute PBGs. Keywords: Absolute photonic band gap; Silicon annular photonic crystal; Anisotropic tellurium 1. Introduction Within the past decades, there has been great interest in the study of PCs [1,2], both theoretically and experimentally, due to their unique ability to control the flow of light at optical wavelength scale [3]. PCs, periodic dielectric materials with a photonic band gap, enable one to control the optical emission properties of materials placed inside them. Many of the promising applications of two- and three- dimensional PCs depend on the location and width of PBGs [4-9]. Although three-dimensional (3D) PCs suggest the most interesting idea for novel applications, two-dimensional (2D) PCs have also been strongly studied, since they can be fabricated more easily than 3D ones and may be employed in optical and electronic devices [10]. Moreover, 2D structures were widely utilized in light emitting 1
diodes, low-threshold lasers, all-optical switches, and high-Q microcavities [11-15]. Therefore, significant attention has been drawn to design and fabrication of 2D PCs. The modes in 2D PCs can be classified as transverse-electric (TE) or transverse-magnetic (TM). An absolute PBG exists for PC only when PBGs are present in both polarizations and overlap with each other. Obtaining a large absolute PBG is very important for various applications [14]. Therefore, it is significant to study the characteristics of the PBG and design the structure with PBG as wide as possible. To our knowledge, employing an anisotropic material or magnetic material, structural deformation, composite structures and changing the shape of scatterer are all the approaches to obtain wide PBG [16-19]. Recently, it has been reported that the suggested class of the so-called annular PCs, where dielectric rods are embedded into air holes of larger radius, have show large PBGs [11]. The designed structure is relatively simple and effective to control PC properties for different polarization. Then, there are many studies on the annular PCs. Most of the works analyze the structure with circle rods embedded into circle air holes. However, if we introduce the noncircular scatterers into the annular PCs, will change the photonic band structure, and may obtain a large PBG. Moreover, silicon PCs are now attracting much attention. A class of integrated optical devices is built in the platform of the structure [20-22]. Especially, the creation of large absolute PBGs in certain frequency regions is an important issue for the design of most optical devices. Therefore, we pay more attention to the 2D silicon annular PCs, and hope to find the effective methods to obtain largest absolute PBGs. In this paper, we investigate quasi 2D triangular annular PCs with square and hexagonal inner scatterers. First, using the plane wave expansion (PWE) method, we compute the absolute PBGs for Si-based and Ge-based annular PCs. The results reveal that silicon annular PCs, regardless of circular, square, or hexagonal inner rods, are unable to possess a larger absolute gap than triangular air-hole PCs, and have small gap size. However, Ge-based annular PCs will present larger absolute gaps than the air-hole PCs in certain conditions. Moreover, the results indicate that the annular 2
PC with hexagonal inner scatterers is the optimal structure. Secondly, the effect of inner scatterer orientation on the PBGs is investigated. The result shows the optimal angle value. Finally, we find that the gap significantly increase by using high refractive index or Te rods instead of Si rods in silicon annular PCs. 2. Theory and model In order to determine absolute PBGs in periodic photonic crystal structures, we start with the wave equation for the magnetic field. In inhomogeneous dielectric materials, Maxwell’ equations for the magnetic field can be written in frequency domain as [14,23] 1 2 H(r) 2 H(r) (r) c
(1)
where is the frequency and c is the speed of light. Because of the relation H(r) 0 , H( r ) is transverse. For periodic systems, the magnetic field can be
expressed as a sum of plane waves [24]: 2
H(r) hGei (k+G)r hG, e ei (k+G)r G
(2)
G 1
where k is a wave vector in the first Brillouin zone, G is a reciprocal lattice vector and e ( =1,2) are unit vectors perpendicular to k G . So Eq. (1) is expressed as a linear matrix equation for the dispersion of EM waves [24-26]:
hG,1 2 hG,1 H 2 G,G G hG,2 c hG,2
(3)
where
e e e 2 e 1 H G,G k+G k+G 1 (G-G) 2 2 e1 e 2 e 1e 1
(4)
1( G) is the Fourier transform of the inverse of (r ) . The quantities 1( G) play an important role in the determination of the photonic band structure for both polarizations. We can write
1 (G)
1 1 (r )e-iGr d 2r cell 3
(5)
where is the area of the unit cell. In a 2D photonic crystal， k+G is in the xy plane for all G ’s and we can choose e1 and e 2 so that e1 is parallel to the z axis and e 2 is in the xy plane. In this case, e2 e1 = e1 e2 =0 and the matrix equation (4) splits into two uncoupled scalar equations. This gives rise to two polarizations of the Bloch waves. In the case of E-polarization ( E(r) is parallel to the z axis), hG,1 0 for all G and we have
k+G k+G 1 (G-G)hG, 2 G
2 c2
hG, 2
(6)
For the H-polarization ( H(r) is in the z direction), hG, 2 0 for all G and we have
(k+G) (k+G) G
1
(G-G)hG,1
2 c2
hG,1
(7)
The photonic band structure can be obtained by solving Eqs. (6) and (7). In this paper, we have considered two kinds of quasi triangular annular PCs with square and hexagonal inner scatterers, respectively. The corresponding models are shown in Fig. 1(a) and (b). The half-side length of square and the side length of hexagon are both expressed as r . The scatterers with the refractive index n are inserted in the middle of the air holes with radius R . The refractive index of the dielectric background is n , and a is the lattice constant. The conventional annular PCs with circle inner rods (radius r ) are also shown in Fig. 1(c). The orientation of noncircular rods relative to the lattice axis is defined by the angle , as shown in Fig. 1(d). In 2D PCs, we always decompose the electromagnetic modes into two types. TE modes have H normal to the plane and E in the plane. TM modes have just the reverse: E normal to the plane and H in the plane. The band structures for TE and TM modes can be completely different. It is possible that there are photonic band gaps for one polarization but not for the other [5]. An absolute PBG exists for a 2D photonic crystal only when PBGs in both polarization modes are present and overlap [14].
4
3. Results and discussion In this study, we consider quasi 2D triangular annular PCs, as shown in Fig. 1. First, we analyze the case of the material: Si-Si. The first material designates the background material, and the second one is for the rods. Our purpose here is to study the modification of the absolute PBG by the three geometrical parameters ( R , r , and
). Furthermore, we want to compare the absolute PBG width of three annular PCs with circular, square and hexagonal inner rods. This process is repeated for another case of Ge-Ge, considering the effects of the refractive index on the PBGs. First, we focus on the silicon annular PCs, and that is the case of Si-Si. The refractive index of Si is 3.45. Three annular PCs with circular, square and hexagonal dielectric rods are involved. The absolute PBGs as a function of the inner rod size r for several discrete values of the air-hole radius R are presented in Fig. 2. The relative width of the band gap is expressed as the gap-midgap ratio / m , where
m is the central frequency of the band gap and is the frequency width. The curves of three different models show that, for a fixed value of R and when the rod size r increases, the absolute PBG size first decreases and then oscillates attaining one maximum that is smaller than the starting PBG value. We know that, when r 0 , the model is, in fact, the usual air-hole PCs with a triangular lattice. The result reveals that the Si-Si structure, regardless of circular, square, or hexagonal inner rods, are unable to possess a larger absolute gap than triangular air-hole PCs. Moreover, it can be seen from Fig. 2 that annular PCs with circular rods show slightly wider PBGs than those with square rods. However, in contrast, annular PCs with hexagonal rods would be the optimal silicon annular PC structure to obtain large PBGs. Next, we investigate the PBG of a triangular annular PC with the case of Ge-Ge. The refractive index is 4.0 for Ge. Similarly to the case of Si-Si, here we study the effect of rod shape on the absolute PBG. The results are shown in Fig. 3. It indicates that, with the fixed R and increasing r , the absolute PBG size of three annular PCs first decreases and then oscillates attaining another maximum. However, the obvious 5
difference to Fig. 2 is that in this case the second maximum significantly increases and will higher than the starting PBG value in certain conditions. That is to say, for a certain value of R , Ge-based annular PCs will possess larger absolute gaps than the air-hole PCs. Additionally, we know that if r is too small, PCs will be harder to fabricate. Therefore, we should pay more attention to the second maximum with larger value of r . It can be seen from Fig. 3 that the annular PC with hexagonal inner scatterers is the optimal structure because it presents obviously wider PBGs than the other PC structures when r is larger than 0.1a. And the largest absolute gap of 21.84% is obtained with R = 0.47a and r =0.15a. Now, the effect of rod rotation ( ) on PBG is investigated for quasi triangular Ge-based annular PCs. First, we discuss gap width versus rotation of square inner rods. The orientation of square rods relative to the lattice axis is shown in Fig. 1(d). We fixed R =0.47a and r =0.12a. Square meets the four rotational symmetry operation, and rotation period is 90。. The rotation angle increase from 0。 to 45。, the rotating step size is 5。, and the result is shown in Fig. 4(a). The figure shows that the largest PBG presents at 0。. The absolute PBG size will slowly decrease with the increasing . And then we analyze gap width versus rotation of hexagonal inner rods. We fixed R =0.47a, r =0.15a, and set the orientation angle of Fig. 1(b) to 0。. Hexagon meets the six rotational symmetry operation, and rotation period is 60。. Fig. 4(b) shows that the largest PBG presents at 0。. When increases in a period, the absolute PBG size first decreases and then increases. The gap size will be smallest at the half period ( 30。). In summary, the triangular annular PC as shown in Fig. 1(b) is the optimal structure of the involved models. However, for Si-based annular PCs, in order to obtain larger absolute PBGs, we should take other special measures. Here, we keep the silicon background ( n =3.45), and involve hexagonal inner rods with different refractive index n . We analyze absolute PBGs variation with inner rod size, when
6
n increases from 2 to 6 with the increment of 1. The air-hole radius value R is 0.45a .
Shown in Fig. 5, when n is less than 4, the absolute gap will decrease to zero with the increasing r . However, if the refractive index increases from 4 to 6, the absolute gap width first decreases and then oscillates attaining another maximum, which is obviously higher than the starting PBG value. Moreover, compared with silicon annular PCs of Fig. 2, we indicate that introducing high refractive index inner rods in silicon background can effectively increase the absolute gap. When n =6, absolute gap reaches to 16.37% as r = 0.08 a . Furthermore, we involve the anisotropic material Te in the PCs, consider the case of Si-Te. The anisotropic Te has two different principal refractive indexes: the ordinary refractive index no 4.8 and the extraordinary refractive index ne 6.2 . Fig. 6 illustrates absolute PBG as a function of r . It can be seen from Fig. 6, the Si-Te structure present larger PBGs than those of the conventional air-hole PCs. Compared with Fig. 2(c), we find that the gap significantly increase by involving the Te rods. The largest absolute gap of 18.43% is obtained when R =0.45a and
r =0.07a. As a result, involving high refractive index or anisotropic inner rods in silicon background, is an effective method to obtain largest absolute PBGs for silicon PCs. 4. Conclusions In conclusion, we analyze 2D triangular annular PCs with square, hexagonal and circular inner scatterers. The effects of structural parameters on absolute PBGs have been investigated. The study indicates that the optimal inner scatterer is hexagonal. Considering the fabrication of PCs, we present the optimum inner rod size, air-hole radius, and rotation angle. Specially, for silicon annular PCs, absolute PBGs significantly increase by using high refractive index or anisotropic Te rods instead of Si rods. The creation of large PBGs in certain frequency regions should be helpful in designing most optical devices. Acknowledgements
7
This work was supported by the National Natural Science Foundation of China (Grant No. 11504100), and the fund for excellent youths of Hubei Provincial Department of Education (Grant No. Q20153004). References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486. [3] M. Turduev, I. H. Giden, H. Kurt, J. Opt. Soc. Am. B 29 (2012) 1589. [4] H. Kurt, D. Yilmaz, A.E. Akosman, Opt. Express 20 (2012) 20635. [5] J.D. Joannopoulos, R.D. Mead, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, 1995. [6] H. Wu, D.S. Citrin, L.Y. Jiang, Appl. Phys. Lett. 102 (2013) 141112. [7] J.D. Cox, M.R. Singh, G. Gumbs, M.A. Anton, F. Carreno, Phys. Rev. B 86 (2012) 125452. [8] L. Feng, M. Ayache, J. Huang, Science 333 (2011) 729. [9] A. Hatef, M.R. Singh, Phys. Rev. A 81 (2010) 063816. [10] T. F. Khalkhali, B. Rezaei, M. Kalafi, Opt. Commun. 284 (2011) 3315. [11] H. Kurt, D. S. Citrin, Opt. Express 13 (2005) 10316. [12] D. Liu, Y. H. Gao, D. S. Gao, X. Y. Han, Opt. Commun. 285 (2012) 1988. [13] S. J. An, J. H. Chae, G. -C. Yi, G. H. Park, Appl. Phys. Lett. 92 (2008) 121108. [14] D. Liu, Y. H. Gao, Aihong Tong, Sen Hu, Phys. Lett. A 379 (2015) 214. [15] Y. H. Park, Y. H. Shin, S. J. Noh, Y. M. Kim, S. S. Lee, C. G. Kim, K.S. An, C.Y. Park, Appl. Phys. Lett. 91 (2007) 012102. [16] Z. Y. Li, L. L. Lin, B. Y. Gu, G. Z. Yang, Physica B 279 (2000) 159. [17] B. Rezaei, T. F. Khalkhali, A. S. Vala, M. Kalafi, Opt. Commun. 282 (2009) 2861. [18] J. Hou, D. S. Citrin, H. Wu, Opt. Lett. 36 (2011) 2263. [19] T. F. Khalkhali, B. Rezaei, A. S. Vala, M. Kalafi, Appl. Opt. 52 (2013) 3745. [20] L. Gan, C.Z. Zhou, C. Wang, R.-J. Liu, D.-Z. Zhang, Z.-Y. Li, Phys. Status Solidi A 207 (2010) 2715. [21] Y.L. Fu, X.Y. Hu, Q.H. Gong, Phys. Lett. A 377 (2013) 329. 8
[22] J.F. Tao, J. Wu, H. Cai, Q.X. Zhang, J.M. Tsai, J.T. Lin, A.Q. Liu, Appl. Phys. Lett. 100 (2012) 113104. [23] D. Liu, H. Liu, Y. H. Gao, Solid State Commun. 172 (2013) 10. [24] T. Pan, F. Zhuang, Z.Y. Li, Solid State Commun. 129 (2004) 501. [25] K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65 (1990) 3152. [26] M. Plihal, A.A. Maradudin, Phys. Rev. B 44 (1991) 8565.
9
Figures Caption Fig.1. The 2D triangular annular PCs with different inner scatterer shapes: (a) Square; (b) Hexagonal; (c) Circular. (d) Rotated inner rods with angle . Fig.2. The absolute PBG variation as the inner rod size for Si-based annular PCs with different inner scatterer shapes: (a) Circular; (b) Square; (c) Hexagonal. Fig.3. The absolute PBG variation as the inner rod size for Ge-based annular PCs with different inner scatterer shapes: (a) Circular; (b) Square; (c) Hexagonal. Fig.4. Gap width versus (a) rotation angle of square inner rods for R =0.47a and r =0.12a, and (b) rotation angle of hexagonal inner rods for R =0.47a and r =0.15a in Ge-based annular PCs. Fig.5. The absolute PBG variation as the hexagonal inner rod size for R = 0.45a and n =3.45. Fig.6. The absolute PBG variation as the hexagonal inner rod size for Si-Te structures.
10
(a)
(b)
(c)
Fig. 1
11
(d)
(a)
(b)
(c) Fig. 2
12
(a)
(b)
(c) Fig. 3
13
(a)
(b) Fig. 4
14
Fig. 5
15
Fig. 6
16