1. Introduction

Photonic crystals [1] (PhCs) possess the tremendous capability of controlling the flow of light at the micrometer scale. At the present time, a part of the photonics community focuses its research on the integration of two- and three-dimensional (2D-3D) photonic devices into photonic chips [2], aiming at the possibility of both emitting and controlling the propagation of light within a single device. Consequently, linear PhC waveguides [3,4] have been of great interest, because they enable the accurate control of light propagation through photonic structures and therefore represent a significant step toward the realization of efficient all-optical technologies. While progress is constantly being made theoretically and experimentally on the creation of well-defined defects in 3D PhCs [5,6], the design of proper broadband single-mode waveguides remains a challenge. Recently, a novel approach for the design of such waveguides has been proposed by Chutinan et al. [7]. They introduced the concept of hybrid 2D-3D photonic band gap (PBG) heterostructures, which consists of a 2D PhC layer intercalated between two 3D PBG PhCs. In this way, we expect the latter to confine light more efficiently to the plane than traditional index guiding confinements. A large variety of 3D structures with omnidirectional PBGs have been studied in the last several years [8], yielding a quite consequent number of candidates that could be used as the 3D PhC cladding of a heterostructure. Silicon inverse opals [9,10], that are known as a cost-effective way for creating photonic devices, meet the necessary requirements for this task. Simple linear defects in 3D inverse opals have recently been investigated [11]. In comparison to this approach, 2D-3D heterostructures offer extra parameters that can be tuned to improve the photonic properties of the structure. Single-mode waveguiding has been demonstrated in inverse opal-based heterostructures with a maximal waveguiding bandwidth of 74 nm centered on 1.55 μm, which seems rather small as compared to the bandwidths obtained with other 3D PBG structures (up to about 180 nm) [12]. In this paper, we propose an improvement of the inverse opal-based heterostructure mentioned above in terms of efficiency and experimental flexibility. We show that a change in the lattice of the 2D layer can yield single-mode waveguiding bandwidths more than 70 % larger than the last reported results, while exhibiting a lower dependence on the 2D layer parameters.

2. Design of the structure

For the sake of comparison, the 3D PhC cladding that we consider is the optimized inverse opal structure that was used by Chutinan et al. [12]. In order to insure the confinement of light to the 2D PhC, the 3D PBG has to be omnidirectional. Partial 3D PBGs cannot be used because the projection of their bands onto the Brillouin zone of the 2D PhC would necessarily result in the appearance of bands into the 2D PBG range of frequency. For this reason, the partial ΓL PBG of the inverse opal cannot be exploited for waveguiding purposes. Lousse et al. showed that this partial PBG could also not be used in the case of simple linear defects [11]. The inverse opal structure consists of a compact face-centered cubic (FCC) lattice of air spheres of radius 0.5a, where a is the distance between the center of two adjacent spheres, surrounded by silicon spherical shells of radius 0.625a and refractive index n=3.45. The spheres are connected by cylindrical windows of radius 0.2a. The omnidirectional PBG is centered on a reduced frequency a/λ=0.62 and has a gap-to-mid-gap ratio of 12 %.

The dielectric rods of the 2D layer have to be placed on the dielectric regions of the 3D inverse opals requiring the periodicity of both PhCs to be similar. In the waveguide configuration that was initially proposed by Chutinan [12], the 2D layer consisted of a triangular arrangement of silicon rods in air, in which an air waveguide was created by removing one row of rods along the ΓK direction of the triangular lattice. In order to stay close to the experiment, the 2D layer was placed on the (111) plane of the FCC lattice, on which opals naturally stack. Figure 1 shows two cross sections of the inverse opal structure at different heights along its growing direction. The inclusion of a 2D triangular lattice of rods could be realized at the position shown on Fig. 1(b), which corresponds to a cut halfway between two layers of spheres. However, for periodic arrangements of dielectric rods in lower refractive index backgrounds, it is known that the graphite lattice exhibits larger PBGs than the triangular one [1,13]. For this reason, we expect a larger on-chip 3D PBG, i.e. the PBG that remains after the inclusion of the 2D layer modes within the 3D inverse opal PBG, so that a larger spectral range would be available for full confinement. The position on Fig. 1(c), which corresponds to a cut in the middle of a layer of spheres, exhibits a graphite pattern of dielectric medium and thus enables the inclusion of a graphite lattice of rods. An air waveguide can be realized by removing one row of rods along the ΓK direction of this structure. Figure 2 is a schematic of the heterostructure obtained.

 

Fig. 1. Schematic of an inverse opal structure stacked along the <111> direction of its FCC lattice. (a) is a side view of the inverse opal. (b) and (c) are cross-sections of an optimized inverse opal structure halfway between two layers of spheres and in the middle of one, respectively. In (b) and (c), the black and white areas correspond to silicon and air regions, respectively.

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Fig. 2. Schematic of an inverse opal-based 2D-3D heterostructure with a 2D PhC layer, consisting of a graphite arrangement of dielectric rods in air. The 2D layer parameters correspond to the structure yielding maximal waveguiding bandwidth (rod radius r=0.10a and layer thickness t=0.20a).

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3. Modelling and discussion

In order to simultaneously obtain in- and out-of-plane confinements from the 2D layer and from the 3D inverse opal claddings, respectively, structure parameters have to be optimized. Since the periodicity of the 2D layer and the refractive index of the rods are fixed by the inverse opal parameters, this optimization can be made by scanning the radius r of the rods and the thickness t of the layer as parameters of the heterostructure and by calculating the photonic band structure for each case. Fully-vectorial eigenmodes of Maxwell′s equations with periodic boundary conditions were computed by preconditioned conjugate-gradient minimization of the block Rayleigh quotient in a planewave basis, using a freely available software package [14].

The largest single-mode waveguiding bandwidth is obtained for a rod radius of r=0.10a and a layer thickness of t=0.20a. The dispersion relation and mode profile of this waveguide are shown on Fig. 3. The single-mode ranges from 0.586 to 0.637 in units of reduced frequency (a/λ), corresponding to a bandwidth of 129 nm centered on 1.55 μm. This bandwidth is 74 % larger than the maximal bandwidth reported in the triangular case (74 nm). As shown on Fig. 3, the single-mode spans almost the whole on-chip 3D PBG, which itself occupies a large part of the inverse opal 3D PBG. For both the triangular and the graphite cases, the waveguiding bandwidth is unavoidably limited by the width of the on-chip 3D PBG. While the optimization of the structure parameters insures that the single-mode spans most of the available on-chip 3D PBG, the graphite lattice yields a larger spectral range for the single-mode to spread on, which explains the significant enlargement of the waveguiding bandwidth.

 

Fig. 3. Dispersion relation of the optimal single-mode waveguide along the ΓK direction of the graphite lattice. The inset shows the time-averaged magnetic-field energy density of the single-mode at the wavevector ka/2π=0.2.

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A large bandwidth has however no consistency if it can be obtained only for very specific values of rods radii and layer thicknesses. It is therefore worthwhile to evaluate the stability of the bandwidth over those parameters. The single-mode waveguiding bandwidth is plotted with respect to the rod radius and layer thickness in Fig. 4. It exhibits a very low dependence on both layer parameters. In particular, bandwidths larger than the maximal results obtained in the triangular case (74 nm) can be found almost on the whole range shown as well as bandwidths above 100 nm for layer thicknesses between 0.12a and 0.24a and rods radii between 0.08a and 0.16a. From an experimental point of view, this gives an important degree of freedom on the design of the 2D layers in the sense that structure parameters may be chosen with more flexibility according to the experimental means employed and that inaccuracies in the elaboration process of the layer may not necessarily destroy the resulting bandwidth.

 

Fig. 4. Single-mode waveguiding bandwidth as a function of the layer thickness and rod radius. The white lines drawn on the bandwidth surface are isosurface curves delimiting bandwidths from 40 nm to 120 nm in steps of 10 nm.

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Finally, in order to demonstrate the robustness of this design and since the elaboration of the 3D inverse opal may result in non-optimized structures, we propose to investigate similar single-mode waveguiding issues in non-optimized, fully infiltrated, inverse opals. The 3D structure consists of close-packed air spheres of radius 0.5a in a complete silicon background. The 3D PBG is 4.5 % wide, centered on a reduced frequency a/λ=0.572. A maximal single-mode waveguiding bandwidth of 58 nm centered on 1.55 μm is obtained with a graphite lattice of rods with radius r=0.15a and layer thickness t=0.15a. This bandwidth is 81 % larger than its triangular counterpart (32 nm). This additional result clearly shows the significant advantage of graphite lattices of rods over triangular ones and suggests that the use of such lattices can be extended to all types of inverse opal structures as long as they possess an omnidirectional PBG.

4. Conclusion

In summary, we have demonstrated that inverse opal-based 2D-3D heterostructures using graphite lattices of rods in air as their 2D layer are much more reliable in terms of single-mode waveguiding bandwidth and experimental flexibility than those using triangular lattices of rods. In particular, we have shown that it is possible to increase the maximal single-mode waveguiding bandwidth by more than 70 %, up to 129 nm centered on 1.55 μm, and to lower the sensibility to the 2D layer parameters. These results are important considering the recent advances in the fabrication of well-defined defects in inverse opal-based photonic structures. We hope that our design will offer new opportunities for the fabrication of single-mode waveguides with large bandwidths in 2D-3D heterostructures.