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We present a procedure for optimizing two-dimensional (2D) Photonic Band Gap (PBG) structures. The procedure discretizes the unit cell of a PBG structure into a binary cell and uses Direct Binary Search to search through a terrain of possible solutions in order to find a more optimal one. This process is designed either for improving the absolute band gap or opening a new one, for a predefined PBG structure. By applying the procedure on a honeycomb array of high dielectric objects in an air background, we increased its Maximum Absolute Gap-to-Midgap Ratio (MAGTMR) to more than twice that of the initial structure. To further prove the utility of this procedure, we also present other examples.
©2003 Optical Society of America
1. Introduction
Photonic band gap structures, or photonic crystals[1–3], have attracted a tremendous amount of attention in the last two decades due to their potential application to Photonic Integrated Circuits (PICs) [4–6], zero-threshold lasers [7], telecommunication devices [8, 9], antireflection coatings[10], and polarizers [10]. Most of these applications depend on the existence of an absolute photonic band gap, by which we mean a stop band that exists in all possible propagation directions within the lattice and for both polarizations. The initial goal towards the true realization of photonic crystals (PhCs) is to search for structures that posses a wide absolute band gap, which can also be used in a broad range of additional applications, such as optical mirrors [11, 12], limiters [13], filters [14] and switches [15]. So far, it has been found that the band gap of PhCs is controlled by the dielectric contrast [2, 16, 17] between the host material and the constituent objects, lattice type [1, 18–22], filling factor [2, 23] (area/volume percentage occupied by the constituent objects), and the shape and orientation of constituent objects [18, 23–29]. Since the first demonstration of an absolute PBG in a diamond-like structure, at microwave frequencies [20], significant effort has been put into seeking alternate PhC structures. To this end, Meade et al. [30], and Ho et al. [31] were the first to report a hexagonal lattice of air holes that contained an absolute band gap. Later, D. Cassagne et al. [19] showed that a honeycomb lattice of GaAs cylinders in air also exhibited an absolute band gap. Subsequent to this work, there have been many approaches proposed for increasing the size of the absolute band gap, or even opening a new band gap within the dispersion diagram. Zhiyuan Li et al. [16, 17] showed that an anisotropic medium can be used to greatly increase the size of the absolute photonic band gap for both square and hexagonal lattices. According to this work, the anisotropic refractive indices of the medium could be tuned to match band gaps for both TE (H-field is directed along the axis of the constituent objects) and TM (E-field is directed along the axis the constituent objects), thus producing an absolute band gap. By this method, they improve the MAGTMR to 14.8% for a square lattice and 17.9% for a hexagonal lattice of Te cylinders in air, respectively. However, the optimal absolute band gap obtained for two-Dimensional (2-D) PBG structures by using this technique is difficult to achieve because it requires a very strong anisotropic material, which is hard to find in naturally occurring materials. Cheryl M. Anderson and Konstantinos P. Giapis [27, 28] increased the absolute band gap for some structures as well by reducing the lattice symmetry to break the band degeneracies in order to open up an absolute band gap. They accomplished this either by inserting a small rod in the center of each unit cell or by interpenetrating two lattices with rods of different size. With the same idea of lifting the band degeneracies by symmetry reduction, others [18, 24, 25, 29] have found structures with a larger absolute band gap by changing the shape and/or the orientation of the constituent objects. These ideas provide insights into designing structures with an absolute band gap. However, because the relationship between the degree of symmetry reduction and the size of the absolute band gap is not known apriori, researchers must painstakingly try dozens of structures in order to ensure that all possibilities have been considered. Aside from being a very time-consuming process, it is possible that there exist some unknown factors that prevent people from creating the optimal structure. To this end, we propose a design procedure that comprehensively takes most possible factors affecting the absolute band gap of a structure into consideration to optimize the PBG structure. To show the capability of this procedure, we apply it to optimize the three most common structures found in the literature: hexagonal, honeycomb, and square lattices, in terms of MAGTMR. The details of this method are presented below.
2. Band gap optimization
The main idea behind this method is the sampling of the unit cell by a factor of a/17, where a is the lattice constant. The resolution, i.e., the minimum feature size, is actually the spatial step Δ used to discretize the unit cell of a PBG structure before optimization. The dispersion properties of the sampled grid, which contains the initial unit cell, are then determined using the Plane Wave Method (PWM) [32]. Subsequent to analysis, the material properties within the sampled unit cell are systematically changed until either performance criteria, e.g., wider band gap, is obtained or convergence is observed.
Because a photonic structure is typically composed of two materials: a background material, A, and a constituent material, B, the discretized the unit cell of a structure consists of many grids points, where each point is filled either by the material A or material B, as shown in Fig. 1 for the case of a honeycomb lattice with hexagonal high dielectric cylinders in air with a filling factor of 0.143.
It should be noted that in Fig. 1, we use black grids to represent high dielectric materials and white grids to represent air. From this point of view, any arbitrary PBG structure can be parameterized as a binary structure. Consequently, the optimization of this type of structure is very similar to the optimization of Computer Generated Holograms (CGH) [33–35]. Thus, if the total number of grid points for one discretized unit cell is N, the solution space will be 2N. By assuming a diagonal symmetry in one unit cell we can significantly reduce the solution space to the order of 2N/4, where N can be selected according to computational cost and possible application-specific constraints, such as fabrication constraints. In our examples, we choose the spatial step size, i.e., resolution, to be Δ = a/17, which thereby represents the discretization of the unit cell that we will optimize. At this point, we then draw two diagonals of this unit cell to partition it into four parts. Since we already assumed diagonal symmetry for the PBG structures, these four parts are identical. As such, we can take only one part for optimization and obtain the other parts through symmetry relations within the unit cell. Next, we number the grid points, including those crossed by the diagonals. For example, if we take Δ = a/3, the grid points that need to be optimized can be numbered as shown in Fig. 2, for both the hexagonal lattice and the square case.
Fig. 1. The discretized unit cell of a honeycomb lattice structure of dielectric cylinders in air with a resolution a/17
Fig. 2. The unit cells and the numbered grids used for synthesis (a) the triangular/honeycomb lattice case (b) the rectangular lattice case
This binary sequence is then optimized using the Direct Binary Search (DBS) method [34, 35]. Thus, each time one bit of the binary sequence is inverted, a new structure is created. The MAGTMR is then calculated by PWM and compared with the best result obtained so far, to determine whether or not to accept the new structure, and continue the search. This process is repeated until the MAGTMR converges to either the specified value or it reaches a constant value, thereby denoting convergence. It should be noted that, because we assume the diagonal symmetry in the unit cell of a PBG structure in the optimization, the Irreducible Brillouin Zone used for calculating the MAGTMR has to be defined according to this symmetry. Figure 3 shows the corresponding Brillouin zones and irreducible Brillouin zones for both the honeycomb and hexagonal lattices, as well as the square lattice case.
Fig. 3. The brillouin zones and the irreducible brillouin zones for unit cell with diagonal symmetry (a) the rectangular lattice case (b) the triangular lattice case
Where the regions inside the thick black lines represent the Brillouin zones. The shadow regions are the irreducible Brillouin zones, which are defined by GXM for the rectangular lattice case and GKM for the triangular lattice case, the arrows on the thick lines indicate the directions of the sampled wave vector, k⇀. Symmetry constraints can be easily applied in the optimization by changing the grids to inverse and defining the irreducible Brillouin zones for bandgap calculation.
3. Example designs
We first implemented this procedure on a honeycomb lattice of hexagonal GaAs cylinders in air. The initial structure is the honeycomb lattice of hexagonal GaAs (εr = 12.96) cylinders in air, with a filling factor of 0.143. This has been shown to be one of the best honeycomb lattice structures in terms of MAGTMR [25]. To optimize this structure, we choose the resolution Δ=a/17 to discretize the unit cell of the initial structure. Figure 4 shows the MAGTMR convergence verse iteration times for the optimization of this structure.
Fig. 4. The convergence of the maximum absolute bandgap-to-midgap ratio for optimization of a honeycomb lattice of hexagonal GaAs cylinders in air Figure 4 indicates that some
300 iterations are needed in this case. For a structure with diagonal symmetry which we applied in the optimization, each iteration takes about 10 seconds, and in total, 300 iterations need 3000 seconds, which is less than one hour.
The unit cells for both initial and optimized structures and the dispersion diagrams for these two structures are shown in Fig. 5.
Fig. 5. Optimization of a honeycomb lattice of hexagonal GaAs cylinders in air (a) the unit cell of the initial structure and its dispersion diagram (b) the unit cell of the optimized structure and its dispersion diagram
Where the blue solid lines represent TM dispersion curves, the red dotted lines represent TE dispersion curves and the yellow shadow regions represent absolute band gap for both TE and TM modes. The rest paper uses the same representations, unless indicated specifically. For initial structure, the MAGTMR is 5.76%, while for the optimized structure the MAGTMR was numerically measured to be 13.29%. This represents an improvement of more than twice that for the initial structure. As additional examples, we optimized two of the best found hexagonal and square lattice structures, as reported by Wang et al. [25]. One is a hexagonal lattice consisting of hexagonal air holes with the orientation θ = 24° in GaAs and the filling factor ƒ = 0.805. Figure 6 shows the unit cells and the dispersion diagrams for the structures before and after optimization.
Fig. 6. Optimization of a hexagonal lattice of hexagonal air holes with orientation θ = 24≊ in GaAs and the filling factor ƒ = 0.805 (a) the unit cell of the initial structure and its dispersion diagram (b) the unit cell of the optimized structure and its dispersion diagram
In this case, the MAGTMR is 20.71% for the initial structure, and for the optimized structure the MAGTMR is 21.42%.
The other structure is the square lattice of square air holes with the orientation θ = 30° in GaAs with the filling factor ƒ = 0.68. Similarly, applying the procedure presented above on this structure with a resolution of Δ = a/17 increase its MAGTMR from 13.6% to 21.2%. The corresponding results are shown in Fig. 7.
Fig. 7. Optimization of a square lattice of square air holes with orientation θ = 30° in GaAs and the filling factor ƒ = 0.68 (a) the unit cell of the initial structure and its dispersion diagram (b) the unit cell of the optimized structure and its dispersion diagram
4. Conclusion
In conclusion, we have presented an efficient synthesis procedure for improving MAGTMR of 2-D photonic crystal structures. The basic steps of this procedure include: (1) discretizing the unit cell of a PBG structure, (2) choosing the number of grid points for optimization according to the required symmetry in the unit cell and creating a binary sequence for parameterizing the unit cell, (3) applying the DBS method to optimize this binary sequence, by retrieving the PBG structure from the binary sequence and calculating the MAGTMR of the corresponding structure with PWM, and (4) repeating this sequence until optimal performance is achieved. Three examples have been shown to illustrate the utility of the procedure, with one of them showing an improvement of more than 200% in the MAGTMR.
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