1. Introduction

During last decade, photonic bandgap materials have attracted increasing interests of photophysics and optoelectronics communities due to the possibility of controlling light propagation through the materials [1, 2]. More recently, a growing number of research papers have specifically reported the tunability of bandgap in photonic crystals. For example, various techniques of bandgap tuning have been demonstrated by applying electric field [3, 4], temperature [5, 6], and magnetic field [7], as well as by infiltrating liquid crystals [8]. In addition, mechanical strain has also been examined as an alternative for band-structure modifications and, possibly, for tuning the bandgaps in photonic crystals as desired [9–13].

In this paper, we investigate the modification of bandgap in photonic crystals undergoing mechanical deformation. Considered is the 2D triangular photonic crystal consisting of silicon matrix and cylindrical air rods. We employ the moving least-square (MLS) method that has been recently proposed by the authors for the real-space band-structure calculation of photonic bandgap materials. Our previous work has well demonstrated the accuracy and fast converging behavior of the MLS method [14].

Our current analysis of the deformation (or strain) effect on bandgap modification, follows a distinguishing route from others that have previously addressed the strain-tunable photonic bandgap materials. First, using the MLS method, we numerically solve the linear elasticity problems of the unit cell of photonic crystal, in order to find the actual deformation in this highly inhomogeneous material subject to prescribed displacements at the outer boundaries of unit cell. Then, we perform the real-space band-structure calculation using the MLS method as well, which is very useful for modelling arbitrarily distorted interfaces due to the deformation.

Therefore, taking full advantage of the real-space technique, we focus on the realistic modelling of the deformation-induced change in the shape of interface between matrix and inclusion, and consequently on the effect of the distorted interface on bandgap modification. In this respect, our numerical study is different from some results in literatures where the emphases were put only on reconfiguring the lattice structure by mechanical strain without considering the detailed shape change along interface. The deformation studies and band-structure calculations are provided in section 2 and 3 respectively, followed by comprehensive discussions.

2. Deformations of 2D triangular photonic crystals

In mechanics of materials, strains or deformations are conventionally classified into fundamental modes such as pure shear, simple shear, and uniaxial tension, as shown in Fig. 1 where the undeformed square of dashed line is also given for clear illustration [15]. Each deformation mode has its own characteristics. For example, the pure shear stress implies that only the shear stress component does not vanish at every point in the material. However, photonic crystal is the highly inhomogeneous material accompanying abrupt changes in material constants across the interface between matrix and inclusions. In mechanics’ point of view, the internal responses of matrix and inclusions to prescribed boundary conditions, are quite different from each other. For example, silicon matrix deforms in response to the applied load, while air columns do not transfer the load. It is therefore difficult to define the fundamental deformation modes of photonic crystals that are exactly equivalent to those of homogeneous materials illustrated in Fig. 1. Nonetheless, we borrow here the terminology of those three deformation modes, for convenience. We will use them to specify how we impose the corresponding displacement boundary conditions on the outer boundary of unit cell. The terminology can also be justified for inhomogeneous materials because those three modes provide the only means available to guide systematically how strains be applied in practice.

 

Fig. 1. Illustrations of fundamental deformation modes: (a) pure shear, (b) simple shear, and (c) uniaxial tension. Dashed square is the undeformed original shape.

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We consider 2D triangular photonic crystals involving cylindrical air (ε=1.0) rods in silicon (ε=11.56) matrix. The inclusion’s volume fraction of the undeformed unit cell is given f=0.72. It is well known that this triangularly periodic structure of those dielectric materials has complete photonic bandgap. Three deformation modes mentioned above are separately implemented by imposing corresponding displacements (u_x, u_y) along the boundary of unit cell. That is, we first prepare the undeformed parallelogram unit cell as shown in Fig. 2, and then prescribe the displacement boundary conditions according to each deformation mode. In this paper, we restrict our study to relatively small deformation of 3% strain as in [11]. The strain is defined as ε_ij=(∂u_i/∂x_j+∂u_j/∂x_i)/2 where u ₁=u_x, u ₂=u_y, x ₁=x, and x ₂=y in our notation. It is noted that the prescribed boundary condition is in fact imposed on the silicon’s part only (i.e., not on the air) along the outer boundary of unit cell.

Next, we solve numerically the typical governing equation of 2D plane strain linear elasticity to obtain the displacement field at all points in the silicon matrix. Refer to [16] on how to solve linear elasticity problem using the MLS method. Young’s modulus of 185 (GPa) and Poisson’s ratio of 0.26 are here used for silicon. The traction-free boundary condition is imposed on the circular interface between the silicon and the air, while the silicon’s part out of the outer boundary are fixed as prescribed above. In Fig. 2, the computed results under 3% elastic strains corresponding to each of the three deformation modes are given in comparison with the undeformed original unit cell. It is clearly shown that the circular interfaces are somewhat distorted due to deformation. In the following section, we explore how the distorted interfaces contribute to modifying the original band structures of photonic crystal.

3. Results on the band structures of deformed photonic crystals

Based on the deformed unit cells obtained from linear elasticity problem, we here compute the photonic band structures using the MLS method, and investigate the effect of each deformation mode on bandgap modification. Because the method is a real-space technique as stated before, it is suitable for accurately modelling the slightly distorted interfaces caused by deformation. The unit cell is discretized by 925 nodes and the associated 864 integration cells. 3×3 quadrature points for each cell, and [1, x, y] for polynomial basis vector are also employed in the MLS modelling. Detailed description regarding how the MLS method is used for the highly accurate computation of photonic band structures, can be found in our recent work [14].

 

Fig. 2. Undeformed (red) and deformed (blue) unit cells of 2D triangular photonic crystal with cylindrical air rods: (a) pure shear, (b) simple shear, and (c) uniaxial tension. In each mode, corresponding shear or tensile strain of 3% is applied.

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Fig. 3. Schematic diagrams of symmetry points and zones in the reciprocal lattice of undeformed (left) and deformed (right) photonic crystals.

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When computing the band structures of deformed photonic crystal, we do not take into account the stress-induced change of dielectric constant in silicon matrix, because the deformation considered in this paper is small enough to disregard it [17]. As well as the distorted interface, the lattice distortion deviated from the original perfect triangular lattice, is also brought about by deformation. Consequently, the perfect hexagonal symmetry in reciprocal lattice is broken. In computing band structures, we thus employ the quasi-hexagonal symmetry points of deformed photonic crystals, as illustrated in Fig. 3. For each deformation mode, we are able to obtain the exact geometry of each quasi-hexagon subjected to 3% strain. Three different symmetry zones, labelled by 1 to 3 in the figure, are considered in calculating the band structures [11].

Results on the band structures of deformation modes are given in Fig. 4 where the top, middle, and bottom rows are respectively for pure shear, simple shear, and uniaxial tension. In the figure, ω is the frequency of the monochromatic electromagnetic wave, c the speed of light, and a the radius of cylindrical air inclusions. In each row, the band structure of the undeformed photonic crystal is also given in the left column. The other three columns respectively correspond to the three different symmetry zones of deformed reciprocal lattice as depicted in Fig. 3. The two dashed horizontal lines in each row indicate the upper and lower bound of absolute bandgap of the undeformed original photonic crystal. This original bandgap is overlapped for the clear demonstration of bandgap changes due to each deformation mode.

In Fig. 4, it is notable that both pure and simple shear deformations cause little alteration of the absolute bandgap although band structures are slightly modified along the symmetry lines. In fact, the ‘widths’ of the absolute bandgaps are not significantly changed in all the three deformation cases, under this small amount of applied strain. This conclusion is contrary to a reference [11] which used the plane wave expansion method and thus did not consider the details of interface changes due to deformations. Our results are rather similar to those of more recent references although different types of materials were employed [13, 17].

 

Fig. 4. Photonic band structures under pure shear (top), simple shear (middle), and uniaxial tension (bottom). TM and TE modes are in blue and red, respectively. Dashed horizontal lines indicate the bandgap of undeformed original photonic crystal. Insets in top low illustrate the quasi-hexagonal symmetry zones of the deformed photonic crystal.

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On the other hand, the only remarkable modification of bandgap in our study is found in the case of uniaxial tension (i.e., the bottom row) where the absolute bandgap is shifted down without considerable change of its width. In Table 1, we show that this is mainly because of the deformation-induced change of volume fraction. The volume fractions are here computed based on the deformed shapes obtained from linear elasticity problems of the previous section. In the table, the reduction of volume fraction in the case of uniaxial tension, is more significant

Table 1. Correlations between volume fraction and bandgap shift.

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than in both shear cases. Consequently, the bandgap shift is evident in tensile mode only. This type of correlation between volume fraction and bandgap shift can also be found in the gap map for triangular lattice of air columns in medium of ε=11.4, as given in [1] where the bandgap gets shifted down as the volume fraction decreases near f=0.72, without perceptible change of bandgap width. As already seen from Fig. 2, lattice distortions are somewhat greater in shear modes than in uniaxial tension mode. On the other hand the bandgap shift is prominent in the tension only as above. It is most probably because the effects of lattice distortions are compensated by the interface distortions in the shear modes, which is not the case of uniaxial tension. This strongly implies that, when computing band structures of strained photonic crystals, we must carefully consider not only the lattice distortion but also the shape changes of interfaces.

4. Conclusions

We have computed band structures of 2D triangular photonic crystals of silicon matrix and cylindrical air columns which undergoes deformations subject to 3% strain. In our study, the contribution by shear strains to the bandgap modification is not remarkable, while the uniaxial tension produces considerable bandgap shift. We have shown that this difference between deformation modes comes from the changes of volume fraction that are closely related to the distortion of interfaces between the matrix and inclusions. Our results are expected to serve as practical guidance for tuning photonic bandgaps by applying mechanical strain.