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In this paper, the complete photonic bandgap (CPBG) of two-dimensional photonic crystals (PCs), which are formed by a square array of solid or hollow dielectric rods connected with dielectric veins, are numerically investigated using the plane wave expansion method. It is clearly demonstrated how the CPBG evolves as the pattern of veins or the type of rods changes. An optimal structure with an ultralarge CPBG is found, whose CPBG reaches Δω = 0.22374 (2πc/a), which is larger than those reported in literatures. The proposed structure seems to have promising applications due to its ultralarge CPBG and large fabrication tolerance.
©2011 Optical Society of America
1. Introduction
Controlling the flow of light waves is one of the major challenges in modern optics. With optical telecommunication and computing technologies becoming increasingly important, there is an ever-growing need for devices that will be able to control and manipulate light wave signals. Photonic crystals (PCs) are the most promising candidates due to their engineered structures that allow for just such a photonic functionality on the materials level, enabling the complete prohibition or allowance of the propagation of light in certain directions and at certain frequencies [1–3]. PCs are, in this regard, highly attractive because they allow the design and manipulation of their photonic properties based on the so-called “band-structure engineering”. A further attraction of these structures lies in the possibility of dynamically tuning their optical properties, which may allow for the realization of controllable and functional nanophotonic devices.
PCs are artificial optical materials of periodic dielectric structures with a photonic band gap (PBG) in which light emission and propagation are prohibited [4–7]. For frequencies within a complete photonic bandgap CPBG, there can be no propagation of light waves, whatever the polarization and the wave vector. Various novel optical devices are expected through the use of the PBG and the artificially introduced defect states. A large PBG is needed in various applications such as optical waveguide [6], liquid-crystal photonic crystal fiber [8], negative refraction imaging [9], defect mode PC lasers [10], and defect cavities [11]. In addition, a large PBG can suppress harmful spontaneous emissions in defect-mode PC lasers when the spectral width of the spontaneous emission is broad [12]. Since many applications of PCs mentioned above rely on their PBGs, it is essential to design crystal structures with a CPBG as large as possible. The important task of bandgap engineering is to find proper PC structures with large CPBGs and design methods to fabricate them. In the two-dimensional (2-D) PCs with a square lattice of dielectric rods, an electromagnetic wave can be decomposed into the TM- (in plane magnetic field) and TE- (in plane electric field) polarization modes. For a polarization mode, a PC may exhibits frequency regions where electromagnetic waves cannot propagate in any direction. These frequency regions are called PBGs for this polarization mode. A 2-D CPBG exists if the bandgaps for both TE- and TM-polarization modes are present and they overlap with each other. It has shown that a CPBG can be obtained and improved by a proper design of the shape and size of lattice cells [13,14]. For example, for the square lattice of circular columns connected by veins proposed by Qiu and He may further produce ~15% relative PBG [Δω = 0.0685 (2πc/a)] [15].
It is well known that the PBG for the triangular lattice is larger than that of the square lattice [7] whose Brillouin zone is square shaped. In order to make the CPBG of latter larger than that of the former, we will show numerically that the enlargement of CPBGs can be realized in a 2-D PC with square lattice of dielectric rods. In the present work, we investigate numerically the optimal design of a 2-D PC with a square lattice of dielectric-shell rods connected to its nearest neighbors by intersecting veins in GaAs by using plane wave expansion (PWE) method developed in Ref [1–5,7]. The dielectric-shell rod is composed of a dielectric rod with an air-hole drilled into the rod. The reason for choosing this geometry is that the existence of TM (E polarization) PBG favors in a lattice of isolated high-ε regions, and that for TE (H polarization) PBG favors in a connected lattice [7,15]. Based on Ref [15], we may adopt a novel pattern with dielectric-shell cylindrical rods connected by intersecting veins [a square lattice of dielectric rods connected to its nearest neighbors by veins and cross veins, see Fig. 1(c) ]. Our designed PC has a large CPBG and shows how a maximum CPBG is obtained by optimally connecting the dielectric-shell rods with intersecting veins.
2. Designs and results
We have utilized the PWE method to simulate a square configuration of dielectric rods in air. The number of plane wave utilized depends on the simulated structure, from a minimum of 16384 to a maximum of 262144 plane waves [13]. The convergence accuracies of the results are better than 1%. The basic equations and formulation of different PC configurations in this work, which are formed by a square array of solid/hollow dielectric rods connected with/without dielectric veins, are derived using the PWE method as illustrated in several literatures [1–5].
Figure 1 shows the evolution of enlarging the CPBG in a 2-D PC: (a) Dielectric cylindrical rods with radius R, (b) dielectric cylindrical rods connected by veins [15], (c) dielectric cylindrical rods connected by intersecting veins, (d) dielectric-shell cylindrical rods with shell thickness R-r, where r representing the radius of air-hole in cylindrical rods, (e) the same as (d) but connected by veins, (f) the same as (e) but connected by intersecting veins, (g) dielectric-shell cylinder rods with a bar-like air-hole drilled into the cylindrical rods, where the bar-like air-hole consisting of a square with length t and two apex semi-circular with radius r’ as shown in Fig. 1(j), (h) the same as (g) but connected by veins, and (i) the same as (h) but connected by intersecting veins. As shown in Fig. 1(a), the dielectric rods and veins in a square lattice (with a representing lattice constant a, a = 1μm through this work) are made of GaAs of refractive index n = 3.4 and they are embedded in air (n = 1), the dashed lines represent the unit cell boundaries. The CPBG is, therefore, favored in a compromise crystal with high-dielectric islands connected by narrow veins [see Fig. 1(b)] [15]. Attempts to enlarge the CPBG, we may adopt a pattern with dielectric cylindrical rods connected by intersecting veins [see Fig. 1(c), Fig. 1(f) and Fig. 1(i)].
By using PWE method and scanning R/a, the gap map of Fig. 1(a) is depicted in Fig. 2(a) and exhibits most of the bandgaps of the TM mode. According to the simulation results by Qiu and He (see Fig. 4 of ref. 15), maximum CPBG [Δω = 0.0685 (2πc/a)] of a 2-D square lattice with circular dielectric columns connected by veins occurs as R = 310 nm and the half-vein width d = 38 nm. Based on our simulations, Fig. 2(b) depicts the gap map of Fig. 1(b) by scanning the ratio of d/a with fixing R = 0.280a. The maximum CPBG, Δω = 0.06326 (2πc/a), occurs as d = 0.021a (21 nm). By scanning the ratio of R/a with fixing d = 21 nm, the maximum CPBG can be increased to a value of Δω = 0.0665 (2πc/a) as R = 285 nm [see Fig. 2(c)]. The increment of Δω is due to the veins effects on PBG. In order to observe more veins effects on this structure, we may adopt a pattern with dielectric cylindrical rods connected by intersecting veins [see Fig. 1(c)]. By scanning the ratio of R/a with fixing d = 60 nm, a dramatically high value of Δω = 0.10488 (2πc/a) can be obtained as shown in Fig. 2(d) which is about 43% higher than that of Δω obtained from Fig. 2(c).
Fig. 2 (a) and (c): Gap maps for the structures of Figs. 1(a) and (b) as a function of R/a. (b) and (d): Gap maps for the structures of Figs. 1 (b) and (d) as a function of d/a.
To further improve the width of CPBG (Δω) of the proposed structure, we may adopt a novel pattern with dielectric cylindrical rods connected by intersecting veins and an air-hole with diameter r drilled into the dielectric cylindrical rods in each unit cell as shown in the structures of Figs. 1(d)-(f). Figures 3 (a), (c) and (d) show the gap maps obtained from the structures of Figs. 1(d), (e) and (f) as a function of r/a, and Fig. 3(b) shows the gap map obtained from the structure of Fig. 1(e) as a function of d/a, respectively. Results show that the width of CPBGs for the structure of Fig. 1(d) depicted in Fig. 3(a) is zero (no any CPBG). As shown in Fig. 3(b) (by scanning the ratio of d/a with fixing R = 282 nm and r = 60 nm), the maximum CPGB, Δω = 0.09069 (2πc/a), occurs as d = 20 nm. By scanning the ratio of r/a with fixing R = 282 nm and d = 20 nm, the maximum CPBG can be increased to a value of Δω = 0.09546 (2πc/a) as r = 70 nm. It can be concluded that the holes effects in dielectric rods can also play a key role on enlarging the CPBG in a 2-D PC from the results of Figs. 3(b) and (c). Therefore, it is expected that a larger CPBG can be obtained from the structures of Fig. 1(f) which contains veins and holes.
After scanning the ratio of r/a with fixing R = 300 nm and d = 60 nm, an ultrahigh value of Δω = 0.21644 (2πc/a) is obtained as shown in Fig. 3(d). It can be clearly seen that a distinct enlargement on CPBG is observed by combination the effect of intersecting veins with the effect of air-holes. Compared to the same structure but without air-hole drilled into the dielectric rod as depicted in Figs. 1 (c), the structure of Fig. 1(f) possesses ultralarge CPBG, which is the widest one to our knowledge, and shows 106% and 216.3% the width of CPGB higher than that obtained from Fig. 1(c) and ref [15], respectively.
3. Analysis of fabrication tolerance
Upon implementing a device that takes these designs into account, there are a lot of difficulties to be overcome in fabrication process to drill a perfect circular air-hole in a dielectric rod. In the following simulations, we will explore the influence of non-circular shape and rotational angle of air-hole drilled into dielectric rod on the width of CPBGs. Figures 4(a)-(c) show the gap maps (scanning r’/a with fixing thickness t = 100 nm) corresponding to the structure of Figs. 1(g)-(i), which are the same as the set up in Figs. 1(d)-(f) but replacing a circular air-hole by a bar-like air-hole [see Fig. 1(j)]. Results show that the CPBG for Fig. 1(g) depicted in Fig. 4(a) is nearly zero, for Fig. 1(h) (fixing R = 282 nm and d = 20 nm) depicted in Fig. 4(b) is Δω = 0.0938 (2πc/a), and for Fig. 1(i) (fixing R = 300 nm and d = 60 nm) depicted in Fig. 4(c) is Δω = 0.22374 (2πc/a), respectively. It is worthy to note that the gap maps obtained from Figs. 4(a)-(c) is independent on the shape of air-hole drilled inside the dielectric rod. Due to the more air-hole effect by a bar-like air-hole rather than that of a circular air-hole, the maximum width of CPBG of the structure shown in Fig. 1(i) is higher than that of Fig. 1(f), and shows 242.6% the width of CPGB higher than that obtained from ref. 15.
In the end of this work, we will investigate the influence of the rotational angle θ of the bar-like air-hole in circular column on the width of CPBG. Figures 5(a)-(c) show the normalized frequency as a function of θ corresponding to the structures of Figs. 1(g)-(i). These results show that the CPBG always keeps large as the rotational angle is varied, indicating that the maximum width of CPBGs is also independent on the rotational angle of θ. It means that the width of CPBG can be kept in stable value even the rotational angle of θ varied in the range of 0°-90°. In addition, the simulation results (not shown here) are similar to those of Figs. 4(a)-(c) by varying the air-hole offsets, radii and vein widths, meaning that the proposed structure is robust against manufacturing inaccuracies.
4. Conclusions
We have proposed a 2-D PC structure with an ultralarge CPBG, which is formed by a square lattice of hollow dielectric cylindrical rods connected by intersecting veins. The CPBG of our optimal design reaches Δω = 0.22374 (2πc/a) when the radius of dielectric rod is 300 nm and the half-vein width is 60 nm. There are many published results on enlarging the 2D CPBGs. The maximum CPBG obtained from this work [>0.2 (2πc/a)] is much higher than that [<0.1 (2πc/a)] of representative literatures [1,2,4–7,13–16]. It seems that the proposed CPBG structure may become a favorable candidate of PC due to its ultralarge CPBG and large fabrication tolerance. With the appearance of new methods for fabricating PBG materials [16,17], it is possible for us to fabricate CPBG materials with our proposed structures. Additionally, by using the hollow dielectric rod, this work also provides a new design method based on the rule of thumb for 2-D photonic crystals.
Acknowledgements
The authors acknowledge the financial support from the National Science Council of Taiwan under contracts NSC 99-2112-M-231-001-MY3 and NSC-99-2120-M-002-12.
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