1. Introduction

Since the concepts of photonic crystals (PTCs) and phononic crystals (PNCs) were proposed [1–3], a great deal of work has been devoted to studies on the propagation behaviors of classic waves in these artificial periodic structures. These structures can exhibit photonic bandgaps (PTBGs) or phononic bandgaps (PNBGs) in which the propagation of photons or phonons is prohibited, respectively. In recent years, periodic structures with simultaneous photonic and phononic bandgaps are of interest for the potential applications to new acousto-optical devices and the enhancements of optomechanical [4,5] or acousto-optical interactions. These structures which can control and localize both light and sound in the same spatial region at the same time are termed phoxonic crystals (PXCs) [6] or optomechanical crystals [7]. Many papers have theoretically investigated the PXC structures including one-dimensional (1D) phoxonic cavities [8,9], two-dimensional (2D) [6,10–12] and three-dimensional (3D) PXCs [13,14]. Particular attention has been focused on 1D phoxonic nanobeams (also called phoxonic strips) [15,16], 2D phoxonic slabs with inclusions composed of holes [17,18], pillars [19], “cross” and “snowflake” structures [20] for the reason that these nanostructures could be precisely manufactured by silicon-on-insulator (SOI) technologies.

In this paper, we theoretically describe the design of 3D PXCs with a simple cubic (SC) lattice. Both the phononic and photonic band structures in these PXCs are investigated for a wide range of the geometry parameters. The vibration modes corresponding to PNBG edges are also discussed.

2. Design, methods of calculation and band structures

These PXCs consist of dielectric spheres on the cubic lattice sites connected by thin dielectric cylinders, as shown in the inset panel of Fig. 1(b) . The structural geometry is described by the lattice constant a, sphere radius R and cylinder radius r. PXCs in a SC lattice can be easily and economically fabricated because of the inherent simplicity of the geometry [21–23]. However, up to now only a few works were devoted to the PTBG properties in similar structures [22,23]. To the best of our knowledge, the PNBG properties in such structures have not been reported yet. The phononic band structures in this paper are achieved by the finite element method (FEM) (COMSOL Multiphysics) [24]. In the case of PTCs, we use a homemade finite difference time domain (FDTD) code. The dielectric material (silicon) parameters in the present work are the refractive index n = 3.6, mass density ρ = 2330 kg/m³, transverse and longitudinal wave velocities c_t = 5360 m/s and c_l = 8950 m/s, respectively [10].

 

Fig. 1 Phononic (a) and photonic (b) band structures of the SC lattice PXC with the cylinder radius r = 0.1a and sphere radius R = 0.375a, displaying complete phononic and photonic bandgaps between bands 6 and 7, 5 and 6, respectively. The high-symmetry points follow usual conventions where Γ = (0 0 0), X = (0.5 0 0), M = (0.5 0.5 0) and R = (0.5 0.5 0.5). The inset panel of Fig. 1(b) is the schematic diagram of the 3D SC lattice PXCs consisting of dielectric spheres on the cubic lattice sites connected by thin dielectric cylinders.

Download Full Size | PDF

We first consider a particular example with r = 0.1a and R = 0.375a. Figures 1(a) and 1(b) present the phononic and photonic band structures which are normalized by 2πc/a and 2πc_t/a, respectively, with c being the velocity of light in vacuum and c_t the transverse wave velocity of the dielectric matrix. The corresponding phononic and photonic bandgaps are of the normalized frequency ranges 0.346 < ωa/2πc_t < 0.827 (with the gap-to-midgap ratio of 82%) and 0.406 < ωa/2πc < 0.455 (with the gap-to-midgap ratio of 11.4%), respectively. This particular structure and corresponding bandgaps are dimensionless. In view of telecommunication applications, the PTBG wavelength λ is close to 1550 nm. With this value as the midgap frequency, the lattice constant is a = 667 nm, the sphere and cylinder radii are R = 250 nm and r = 67 nm, respectively. In turn, the midgap frequency of the PNBG is f = 4.7 GHz. This particular example shows that the designed structure indeed exhibits dual bandgaps.

3. Detailed numerical results and discussions

The demonstration of the simultaneous PNBG and PTBG in Fig. 1 necessitates a more detailed investigation of the existence and quality of the bandgaps by varying the geometry of the structure. Figure 2 shows the corresponding phononic and photonic bandgaps as a function of r/a for the fixed sphere radius R = 0.35a. As shown in Fig. 2(a), the maximum PNBG appears with the minimum cylinder radius, and the bandgap widths of two bandgaps (between the 6th and 7th, 12th and 13th bands) generally decrease as the cylinder radius r increases. The upper and lower bandgaps close at r = 0.095a and r = 0.18a, respectively. One can see that the passing band (between 7th and 12th bands) is compressed as r decreases, and ultimately covers a quite narrow frequency range of 0.51 < ωa/2πc_t < 0.52 at r = 0.02a. For the PTBG map shown in Fig. 2(b), the gap opens up at r = 0.07a, and widens up as r increases until r = 0.1a, and then becomes narrower with r increasing. Both the upper and lower bandgap edges decrease with r increasing. The PTBG exists when 0.07 < r/a < 0.2. It can be seen that dual bandgaps appear in the normalized cylinder radius range of 0.07 < r/a < 0.18. The optimal choice of the cylinder radius is r = 0.1a in which case the phononic and photonic bandgaps are over the normalized frequency ranges 0.349 < ωa/2πc_t < 0.763 and 0.418 < ωa/2πc < 0.478, respectively.

 

Fig. 2 Variations of the phononic (a) and photonic (b) bandgaps as functions of the cylinder radius (r/a) with the dielectric sphere radius R = 0.35a. The shadowed areas correspond to the complete bandgaps.

Download Full Size | PDF

For the cylinder radius r = 0.1a, the variations of the PNBG and PTBG as functions of R/a are shown in Fig. 3 . We can observe that there are three PNBGs between the 6th and 7th, 12th and 13th, 15th and 16th bands in Fig. 3(a). Unfortunately only the first bandgap between the 6th and 7th bands is wide, the others are quite narrow and less applicable. So in what follows we only consider the widest one. The PNBG opens up at approximately R = 0.225a at the normalized frequency of 0.4204, and widens up as the sphere radius R increases. The PNBG extends to maximum at R = 0.4a with the gap-to-midgap ratio of 84.5%, and then becomes narrower slightly as R increases. Figure 3(b) shows that the PTBG appears in the similar values of R/a, and the widest gap is found at R = 0.35a. The dual bandgaps appear in a wide normalized sphere radius range of 0.24 < R/a < 0.45.

 

Fig. 3 Variations of the phononic (a) and photonic (b) bandgaps as functions of the sphere radius (R/a) with the dielectric cylinder radius r = 0.1a. The shadowed areas correspond to the complete bandgaps.

Download Full Size | PDF

The above results indicate that there is an optimal geometry which generates both large PNBG and PTBG. For the purpose of obtaining the optimal R/a and r/a, variations of the gap-to-midgap ratio of the first bandgap for the phononic and photonic cases with both geometry parameters R/a and r/a are shown in Figs. 4(a) and 4(b), respectively. For the PNBG map, the gap-to-midgap ratio decreases with r/a increasing. For a given r/a, this bandgap is widest at an intermediate value of R/a and becomes narrower as R/a deviates from this value. The variation of the PTBG with R/a and r/a is more complex. There is a region of the geometry parameters (near R = 0.35a and r = 0.105a) for the widest PTBG; and the gap-to-midgap ratio decreases as the combination of R/a and r/a deviates from these values, see Fig. 4(b). One can see that the simultaneous PNBG and PTBG exist in most area of the geometry parameters in Fig. 4. And in the range of 0.365 < R/a < 0.41 and 0.1 < r/a < 0.125, the gap-to-midgap ratios for the phononic and photonic bandgaps exceed 50% and 8%, respectively. So it is flexible to design the detailed structures according to the applications and fabrication difficulties.

 

Fig. 4 Variations of the gap-to-midgap ratio of the first phononic (a) and photonic (b) bandgaps with the sphere radius R/a and cylinder radius r/a.

Download Full Size | PDF

We also calculate the vibration modes at the edges of the lowest bandgaps (between the 6th and 7th bands), see Fig. 5 . It is noted that the dielectric spheres can be regarded as a periodic arrangement of lumps connected with thin cylinders. For the lower-edge mode (Figs. 5(a)–5(c) correspond to point A and Fig. 5(g) to point C), we can see clearly that this mode can be described by a torsional mass-spring model, i.e., the sphere lump acts as the mass and the cylinders are the torsional springs. While for the upper-edge mode (Figs. 5(e)–5(f) correspond to point B and Fig. 5(h) to point D), the cylinders vibrate and the lumps themselves also vibrate slightly. The effective mass of the lumps and the effective stiffness of the springs increase with the radius of the dielectric spheres increasing, and therefore the eigenfrequencies of the lower-edge decrease and those of the upper-edge modes increase, respectively. The effective stiffness of the torsional springs increases with the cylinder radius increasing, which makes the frequency of the lower-edge increase, as is shown in Fig. 2(a). For the PTC, Biswas et al. [22] have indicated that the dielectric mode (i.e., lower-edge mode) could have a large amplitude at the vertices of the SC lattice, and thus introducing dielectric spheres at these vertices can lower the frequency of the dielectric mode and open up a PTBG.

 

Fig. 5 Vibration modes of the SC lattice PXC at the bandgap edges corresponding to Fig. 1(a). Lower-edge modes (point A) in (a) XY, (b) YZ, (c) ZX cross-sections; upper-edge modes in (d) XY, (e) YZ, (f) ZX cross-sections; panels (g) and (h) correspond to the lower-edge modes (point C) and upper-edge modes (point D) in 3D view, respectively.

Download Full Size | PDF

4. Conclusion

We propose a 3D dielectric PXC with simultaneous large PNBG and PTBG. The pattern of the proposed structure is composed of dielectric spheres arranged in an SC lattice and connected by cylinder rods. The widest PNBG appears with large spheres and thin cylinders; while the widest PTBG appears with an appropriate geometry of the spheres and cylinders. The complete PNBG and PTBG can exist over a wide range of geometry parameters. 3D PXCs are promising structures for integrating the management of sound, light and heat simultaneously with potential applications to acousto-optical devices and highly controllable photon-phonon interactions. And these PXC structures can be suitable for applications such as waveguides and microcavities that exhibit strong optomechanical effects. For instance, after changing the radius of one sphere, novel effects stem from strong simultaneous localization of photons and phonons in the dielectric defect sphere can be anticipated [25]. Finally we mention that under some values of the elastic strain, nonlinear behavior may be revealed in such structures because of the rod connections. This should be an interesting topic.