1. Introduction

Slot-waveguides were originally proposed and experimentally demonstrated by Almeida et al [2, 3, 4]. A slot waveguide is comprised of two high refractive index structures separated by a nanoscale gap (10 to 100 nm) filled with low refractive index material. The guided mode of a slot waveguide has up to 20 times higher intensity than that achieved in conventional rectangular waveguides. This extraordinary increase in the optical power in such a small volume is highly dependent on the index contrast between the two materials. This property has led to several exciting new device concepts [5, 6, 7] and, more recently, the proposal by Barrios and Lipson [8] for a Er⁺-based light-emitting structure in ring-resonator configuration. However, the previous work addresses the vertical configuration of the slot-waveguide. This configuration is not suitable for thin film fabrication techniques. Additionally, it requires high-resolution electron beam lithography to define the very narrow slot structure. Recently Galli, et al, [1] have demonstrated the enhancement of the off-plane photoluminescence emission at 1550 nm in a horizontal slot-like Si/Er⁺-SiO₂/Si structure patterned with a 2D photonic crystal lattice. This configuration uses thin-film deposition techniques to generate the thin (5-30 nm) slot structure, and does not require high-resolution lithography techniques. While ring resonators previously proposed by Barrios and Lipson [8] are also an alternative in this horizontal configuration, they are expected to present higher losses due to the TM nature of the modes. Photonic crystals offer an attractive solution for high-confinement in the horizontal-slot configuration. However, Galli et al, [1] reported that the structure did not present confinement in the vertical direction at that wavelength, which leads to significant losses. In a regular slab photonic crystal waveguide, as well as in resonators, light is confined by a photonic bandgap in-plane, and by total internal reflection (TIR) in the vertical or out-plane direction. Galli’s structure supported only lossy modes at 1550 nm that were inside of the radiation cone of the cladding (light cone). A linear guided mode confined in the slab of such a structure would lead to a Si/Er⁺ -based amplifier structure.

In this paper, we describe a 2D hex slot photonic crystal slab (slot-PhCS) structure, obtained by combining the photonic crystal slab and slot waveguide as shown in the Fig. 1. We investigate two manufacturable configurations for this structure. Slot-like slab modes that lead to confined linear defect modes without radiation loss in vertical direction are determined. In this structure, the slot-like modes are outside the radiation cone, which are not coupled to cladding modes (i.e. modes which are inside the light cone). These results open a new avenue for Er+ light emitting devices on Si substrates.

2. Band-gap optimization for linear modes of slot PhCS

In the following sections, we will analyze the properties of slot PhCS based on silicon slabs (n_Si = 3.46) in two different cladding options that are compatible to current fabrication technologies and present a bandgap. First option of cladding, shown in Fig. 1(a), is air cladding, where we consider air in the holes, the slot layer and in the surrounding cladding. The second option, shown in Fig. 1(b), is an oxide in the slot layer and cladding region, with air holes etched through the structure, similar to Ref. 1. A third option where the air cladding and holes in the first configuration are replaced by a glass or polymer material with index of refraction n=1.46 was considered. This would be desired if one would fill air-holes and cladding with polymer or spin-on glass containing optical gain media, such as quantum dots or Er+ doped nanocrystals. However, this structure cannot support vertical confinement in 1D defect guiding, in-spite of presenting a bandgap for higher modes, and is not presented here.

Simulations of the slot-PhCS structure present a challenge due to the very small width of the gap, in the range of 10-30 nm, which is substantially smaller than typical thickness and period found in slab photonic crystals. Typically, eigenstates of the slot mode and linear defects mode in the slot PhCS are computed using the plane-wave expansion technique for periodic structures, as is done in the analysis methods used in slab photonic crystal [9]. In our case, a super-cell with a triangular in-plane lattice in the slot PhCS and a large cladding, with a period of 4 times the thickness of the slab region, must be used. However, in the out-plane direction, due to memory limitations, the step size is comparable to the thickness of the slot layer.

 

Fig. 1. 2D hexagonal slot PhCS band diagram with two different cladding configurations respectively. a) Refractive index profile of air-slot PhCS structure in the cross-section plane (black line). b) Refractive index profile of air-oxide slot PhCS structure. (c) Band diagram of air-oxide slot PhCS structure (a) for odd-y-parity with silicon slab thickness of 200 nm and slot thickness of 15 nm with r/a=0.37. The inset shows a 3D visualization of the hexagonal slot PhCS. Band gap is about 0.015 a/λ. d) Band diagram of air-oxide slot PhCS structure (b) for odd-y-parity with silicon slab thickness of 310 nm and slot thickness of 15 nm with r/a=0.42. The inset shows a brillouin zone of the hexagonal slot PhCS. Band gap is about 0.015 a/λ

Download Full Size | PDF

The stability and roughness of the resolution parameters were adequate, as shown by only a 0.5% relative bandgap edge shift when doubling the steps in three directions respectively (double resolution). Our calculations were all based on the predetermined slot width (s) of 15 nm as this ensured the presence of a bandgap, while allowing sufficient resolution in the simulations. The computation of the modes in slot PhCS was limited to odd-symmetry solutions, which are pure TM mode in the centered bisecting plane (electric field in the out-plane direction) since this is a requirement for slot-like modes. The odd symmetry condition is

Figure 1(c) shows the simulated band diagram for the air-clad structure shown in Fig. 1(a). The bandgap was maximized first by adjusting the ratio of the radius to the period (r/a) of the holes, followed by adjusting the thickness of the layers of the slab (t) of the slot PhCS. The slot PhCS in Fig. 1(d), with air-holes in an SiO₂ cladding has a similar band diagram and mode properties.

Linear defects are obtained by removing a row of nearest neighbor holes to create a defect waveguide along the ΓK direction. This results in a reduced Brillouin zone inside the original Brillouin zone where the wave vector along the linear defects direction ends at K’ = π/3α instead of the point K = 4π/3α, as previously shown by Johnson et al [9]. We carried out an optimization of the bandgap for the geometry parameters thickness (t/a) and radius (r/a), while ensuring that the upper band edge at the K’ point was outside of the light cone. This is important in order to ensure that the defect mode at the K’ point is outside the light cone.

 

Fig. 2. Mode profile of the first 4 bands showing the normal component of the electric field (y-component) at the center of the slot-PhCS at the K-point. The first row shows Ey in XZ plane with odd symmetry, the second row shows Ey in cross section along the direction marked by the arrow. Note that in the case of (d) the field is zero along the chosen cross section.

Download Full Size | PDF

In air holes and cladding configuration, the maximum band gap is centered at a frequency of 0.48 a/λ, it was achieved for r/a = 0.37 and thickness t = 200 nm. These values imply that a value for the lattice constant a = 744 nm is required to center the bandgap at 1550nm. The resulting bandgap is ∆λ=50 nm wide which is only 12 % of that one would obtain with a TE-like mode bandgap in single slab Si PhCS. Hence the optimization of the bandgap is indeed important. Figure 2 shows the simulated mode profiles for the y-component of the field (E_y) of existing odd symmetric modes in the slot PhCS at the K’ point in the Brillouin-zone. For the odd symmetry, the electric field is strongly confined in the slot layer with about 35% of the total power in the 15 nm slot section. Modes with even symmetry have field distribution similar to that of the conventional slab mode in the y direction. It is important to note that for higher bands the electric field in y direction is still enhanced in the slot, and that additional nodes appear in the xz plane. The slot PhCS in Fig. 1(d), with air-holes in an SiO₂ cladding has a similar band diagram and mode properties. We optimized the bandgap for this structure and found that r/a = 0.42 and t = 310 nm.

3. Guided modes of linear defects in slot PhCS

The benefits of the increased electric field in the slot region can be captured by guided modes. Gain media present on the slot, such as proposed by [1] require such guiding. Similarly, nonlinear processes proportional to E² are greatly amplified by the slot structure. Guided modes in photonic crystals benefit from the slow-wave effect, or dispersion relationship improving further the application of non-linear effects. Properly designed defects in the slot PhCS will confine the field in one (1D) or two-dimensions (2D) in the plane.

 

Fig 3. (a). Dispersion of linear defect waveguide in slot PhCS with 1^st cladding option, with (b) and (c) showing the profile of mode (Ey) in XZ and YX planes respectively. (d) Dispersion of linear defect waveguide in slot PhCS with 2^nd cladding option, with (e) and (f) showing the profile of mode (Ey) in XZ and YX planes respectively.

Download Full Size | PDF

In Fig. 3(a), the guided-mode frequencies for 1D defect waveguides in the air-clad configuration are shown to be inside the bandgap. This was insured by the constraint in the optimization described earlier. Linear defects are simply introduced along the Γ-K direction that a row of holes removed from the pattern. The dispersion relationship of the linear defects waveguide for the second cladding option is shown in Fig. 3(d). Figures 3(b), 3(c) and Figs. 3(e), 3(f) show the electric field in the y-direction for the guided mode of the 1D waveguide for air-clad and air-holes in oxide-cladding. The group velocity of the defect mode waveguide is difficult to estimate due to numerical uncertainty in the calculations, and is approximately ν_g=0.1c for the silica slot structure, which is comparable to other 2D slab waveguides [10]. In comparing this result with those of pure silicon slab photonic crystals, one must notice that the region in k-space where the slot photonic crystal defect mode is inside the bandgap is approximately 15% that of an equivalent slab structure. It is important to note that the high intensity of the electric field in slot layer is kept in the 1D waveguide configuration.

4. Conclusion

We investigated slot photonic crystals with focus on achieving confined 1D guided modes in manufacturable structures. We considered two configurations, an all air-clad structure of air-holes in an air-slot layer, and a SiO₂ clad structure of air-holes in a SiO₂-slot layer. These structures are practical from a manufacturing standpoint, for integration into silicon processing compatible with CMOS processes. The optimum geometrical parameters for slot-slab photonic crystals in Si/air and in Si/SiO₂/air structures that maximize the bandgap for linear guided modes were determined. It is important to note that the bandgap of TM-like modes in slot-PhCS occurs only for modes between the 2^nd and 3^rd bands or higher. The dispersion relationship for the defect waveguide leads to low group velocities and hence increased density of states which can significantly increase the performance of active and nonlinear devices, such as lasers and modulators as described by several researchers in the past few years [10, 11]. These results should guide the design of devices where the slot consists of Er⁺-doped gain media, such as proposed by Galli et al [1]. The confined slab mode and linear defect mode, which have high intensity in the slot layer, can be used in other applications such as modulators, photodetectors, non-linear processes among others. Further applications of these structures are likely to be discovered.