1. Introduction

Photonic crystals (PhCs) [1–3] are a very promising candidate for future telecommunication devices because of their ability of controlling electromagnetic waves. More particularly, a photonic band gap (PBG) [4] may appear for PhCs in some special arrangements, thereby preventing light from propagating through the crystal. The presence of a PBG in PhC structures can give rise to phenomena such as waveguiding, optical switching or optical filtering.

A few years ago, periodic arrangements of dielectric spheres [5–17] arose the possibility of creating ultra-compact and efficient photonic devices at low cost, using techniques such as opal self-organization [5–9]. Micromanipulation of spheres [10] has also been explored to create specific patterns. At present time, high quality bare opals are obtained by using spheres of polymethylmethacrylate (PMMA) or silica (SiO₂), materials that exhibit quite low refractive indices. However, the use of higher refractive index materials could open new possibilities in designing more efficient photonic devices, and toward this aim, a couple of experimental methods have been developed to synthesize higher refractive index spheres. For example, recent studies report the synthesis of titania (TiO₂) spheres with diameters of a few hundreds nanometers with coefficients of variations in size ranging from 5 to 20 % and refractive indices as high as 2.9 [12,13]. One could imagine to use other materials in order to reach high refractive indices.

Recently, monolayers of dielectric spheres received increasing attention [14]. Such structures could be used for example as planar defects in three-dimensional (3D) structures [15] or as building blocks for the creation of 2D/3D hybrid architectures. Moreover, specific patterns could recently be engineered in such monolayers by using micromanipulation [10], electron beam lithography [16] or laser-induced breakdown [17]. These novel techniques, currently adapted to PMMA and SiO₂ spheres in particular, enable the design of new PhC structures that may exhibit very useful properties. One can expect that similar techniques will be adaptable to higher-refractive index spheres in the near future. Considering these recent technological advancements in synthesizing high refractive index spheres and in creating defects in colloidal structures, we focused our work on the study of various sphere-based monolayer configurations in order to find structures that exhibit a PBG, therefore offering lateral confinement, which may be used for example for waveguiding functionalities. In this paper, we present the photonic band structure and gap map of a PhC monolayer where a PBG is found below the light cone and propose a possible waveguide configuration that yields two non-degenerate guided modes.

2. Modelling considerations

Opals arrange themselves into a close-packed face-centered-cubic (FCC) structure, whose dielectric spheres are stacked along the <111> direction, forming a compact triangular lattice. On a modelling point of view, it should be noted that out-of-plane modes do not need to be computed, for they belong to the light cone, continuous region that indicates all the possible frequencies of the radiative bulk background. In-plane modes only need to be computed. It is also recommended to treat the modes according to their even or odd parity with respect to the plane symmetry σ_z in the middle of the structure. In the results given below, the lattice parameter a is taken to be the diameter of the spheres.

We consider that the monolayers are suspended in air, leaving a larger number of vertically confined modes. The presence of a substrate would unavoidably decrease the slope of the light line and as a result, some of the initially confined modes may lie within the light cone and therefore become lossy. However, since this structure has never been studied, the use of an air background is a necessary first step. The dielectric spheres constituting the monolayer have a refractive index of 2.9, which, as shown below, is sufficiently high for gap appearance. Such refractive index corresponds to rutile TiO₂ material, constituting a practical example that can be generalized to all materials whose refractive index lies in that range.

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 [18]. Convergence studies were conducted to estimate the effect of the planewave cut-off and of our use of the supercell approach in our calculations. In the case of spheres of refractive index 2.9, they led to the conclusion that firstly, the use of 16 planewaves per direction and per unit cell was sufficient for an accuracy of better than ± 2 % and secondly, a supercell of 8 unit cells in the vertical direction reduces interaction between adjacent layers resulting in an accuracy of better than ± 1 %.

3. Photonic band structures of monolayers of dielectric spheres

Figure 1 sketches the photonic band structure of a perfect arrangement of dielectric spheres with refractive index 2.9, arranged on the (111) plane of the FCC lattice. No significant PBG is found either in the even or in the odd modes. For both parities, symmetry degeneracies mainly located at the K-point and accidental degeneracies prevent us from obtaining PBGs that are large enough so that they could be used efficiently, for example for waveguiding purposes. Both kinds of degeneracies would not be lifted up by increasing the dielectric contrast of the structure, but they probably would by changing its crystal lattice.

 

Fig. 1. Photonic band structure of a monolayer of dielectric spheres lying on the (1 1 1) plane of a FCC structure. The parity of the modes with respect to the plane of study is indicated by a blue continuous line and a red dashed line for the even and odd modes, respectively. The light cone (dark blue shaded area) is the continuous region were light may couple to the bulk background.

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It has been demonstrated that changes in the unit cell of a 2D hexagonal lattice of cylinders can provide the opening of new PBGs [19]. Based on this approach, we propose a configuration that can be obtained by removing dielectric spheres in a triangular pattern, resulting in a hexagonal superlattice monolayer of spheres, as shown on Fig. 2(a). At present time, such a pattern could be realized by micromanipulation [10], electron beam lithography [16] or laser-induced breakdown [17]. The study of defects in self-organized structures being an active part of the opal-based PhC research, it is conceivable that other fabrication techniques, notably some applicable to higher-refractive index materials, will be developed in the future.

Figure 2(b) sketches the photonic band structure of the even modes of the hexagonal superlattice monolayer of spheres. The PBG that is obtained has a gap to mid-gap ratio of 12.4 % and is centered on a reduced frequency ωa/2πc=0.214, tunable to the telecommunications range (λ=1.55 μm) by using dielectric spheres of diameter Φ=0.33 μm. It should be noted that spheres of such diameters were recently synthesized [12,13].

As it is known, the higher the refractive index of a material, the lower the frequency of its modes. Therefore, by increasing the refractive index of the spheres, we expect to observe modes at lower frequencies. A gap map of the structure is given in Fig. 3 for the even modes. The larger PBG appears between the 1^st and 2^nd bands for dielectric constants greater than 3.7 (i.e. for refractive indices greater than 1.93). It becomes wider and is pushed down in frequency with the increase of the refractive index. As a result, the PBG spans a larger area in reciprocal space, lessening the out-of-plane radiation. A smaller PBG appears between the 4^th and 5^th bands for spheres with dielectric constants greater than 10.4 (i.e. for refractive indices greater than 3.22), but we considered it to be too small to our concerns.

 

Fig. 2. (a) Primitive cell of the hexagonal superlattice. It is composed of four regular hexagonal cells and counts a total number of three spheres per supercell. (b) Photonic band structure in the even modes of a hexagonal superlattice monolayer of dielectric spheres with refractive index 2.9. The guided modes lying below the light cone (dark blue shaded area) exhibit a PBG (light blue shaded area) centered on a reduced frequency ωa/2πc=0.214 and with a gap to mid-gap ratio of 12.4 %, being significant enough for preventing light in a quite large frequency range from propagating through the monolayer.

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Fig. 3. Gap map of a hexagonal superlattice monolayer of dielectric spheres in the even modes. The PBGs (dark blue shaded area) that exist below the light cone are widened and pushed down to lower frequencies as the dielectric constant ε=n ² of the spheres is increased

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We should finally remark that the addition of a substrate below the monolayer may place the confined modes above the light line and despite their induced vertical losses, these modes would still appear as resonances in the radiative continuum. Thus, the effect of the 2D PBG would still remain and offer lateral confinement within the monolayer. Additionally, the mirror symmetry that enables the separation of the modes according to their even or odd parity would be broken. However, as it is the case with 2D PhC slabs on substrates, it would be possible to keep the main features of the modes. With a sufficiently high dielectric contrast between the monolayer and the substrate, the latter would act only as a small perturbation to the system. Highly confined modes would still exist within the monolayer and the effect of the 2D PBG would be sustained.

To summarize, the larger PBG that was found indicates that an incoming EM wave at this range of frequency would not be able to propagate through the monolayer. Such structures can be used for a large number of different applications and researchers can use this phenomenon according to their needs. For example, by mixing the hexagonal superlattice and the standard compact structures, it is possible to confine light in certain regions of the monolayer in order to obtain the functionalities that we can expect from a 2D PBG, i.e. mirrors, cavities, etc., and therefore engineer novel sphere-based devices. 2D/3D hybrid architectures constitute an other possibility. In these structures, a 2D confinement can be used to insert or collect light from a 3D structure. Now, in order to prove the functionality of the hexagonal superlattice monolayer, we present one practical application among many others, i.e. waveguiding, where the inclusion of defects in the structure would create defect modes, confined vertically by index guiding and laterally by the presence of the PBG, if they lie within it.

4. Linear waveguide

The inclusion of defects for waveguiding purposes is not straightforward. Intensive modelling efforts are required to design geometries with suitable waveguide modes lying within the PBG. Since the presented structure could be made by removing spheres in a triangular pattern, a linear defect could be fabricated by leaving some of them along one particular direction. We propose to create a linear defect by leaving one row of spheres along the ΓK direction of the structure mentioned above. On a modelling point of view, since a linear defect is confined in two dimensions, an artificial periodicity in the lateral direction needs to be added to the computational supercell, therefore preventing two adjacent waveguides from interaction. The addition of 5 cells in the lateral direction is sufficient to decouple the waveguides.

Figure 4 sketches the photonic band structure of the even modes along the ΓK′ direction [20]. The inclusion of this linear defect gives rise to two non-degenerate modes with reduced frequencies of ωa/2πc=0.204 and 0.214 at the K′-point. The cross-sections of the vertical component of their magnetic field are shown in Fig. 5. These two defect modes are truly due to the effect of the PBG and are well-confined to the waveguide. Their flatness indicates that their group velocity is very low, and although this may yield to some difficulties in coupling light to these lossless modes into the waveguide, it could be interesting for reaching slow light or for lasing applications. The creation of resonant cavities such as closed linear defects using the hexagonal superlattice structure should therefore be investigated in the future. On the contrary, if one is interested in propagating light into the waveguide, the bandwidth of the waveguide could probably be widened by changing the shape of the defect, its optical properties, or by inserting additional defects.

 

Fig. 4. Photonic band structure of the even modes of a linear waveguide directed along the ΓK′ direction. Modes lying within the light cone (dark blue shaded area) or the slab bands (light blue shaded area) may couple to the bulk background or escape laterally in the 2D PhC, respectively. Two guided modes are found lying within the PBG.

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Fig. 5. Horizontal (top) and vertical (bottom) cross-sections of the vertical component of the magnetic field of the lower (a) and higher (b) non-degenerate guided modes at the K′-point. The waveguide is made by inserting a row of dielectric spheres along the ΓK′ direction. The magnetic field (blue shaded at its minima, red shaded at its maxima and green shaded at its node) is well-confined to the waveguide.

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Finally, it is worth noting that since our structure consists of removing dielectric spheres from a close-packed monolayer according to a triangular pattern, the resulting field patterns are very close to the one obtained with a triangular array of holes in a dielectric slab [20]. In other words, we could define this waveguide as a sphere-based W1 triangular lattice PhC waveguide.

5. Conclusions

In this paper, we demonstrated that a hexagonal superlattice monolayer of dielectric spheres exhibits a photonic band gap in the even modes below the light cone and studied its evolution with respect to the refractive index of the spheres. This gap exists for spheres with refractive indices greater than 1.93. This monolayer configuration could be used for in-plane waveguiding, where the inclusion of a linear defect enabled us to confine light along one direction, thereby creating a theoretically lossless waveguide. This structure could open new possibilities for in-plane waveguiding, for reaching slow light or alternatively be the building block of 2D/3D hybrid architectures, mirrors, cavities and other applications that may find use in the future design of all-optical technologies at low-cost. Considering the recent technological advancements in engineering specific patterns on monolayers of spheres and in synthesizing high refractive index spheres, we expect these theoretical results to be checked experimentally in the near future.