1. Introduction

The slot waveguide structure [1, 2, 3] has attracted considerable attention because it allows for extremely strong confinement of light in a low index material embedded between two silicon wires. This feature is of particular interest for the efficient exploitation of third-order nonlinear effects in low index media. Koos et al. [4] theoretically studied the dependence of the effective area of vertical slot waveguides on the geometry and predicted the feasibility of nonlinearity coefficients of up to 7 × 10⁶ W^-1km^-1. Recently, the same group experimentally demonstrated vertical slot waveguides with a nonlinearity coefficient of 1.04 × 10⁵ W^-1km^-1 [5].

Vertical slot waveguides can be fabricated from a single monocrystalline silicon layer but put high demands on the fabrication process due to the small feature sizes. The minimum achievable slot width is practically limited to >50 nm. Moreover, side wall roughness is a critical issue. Horizontal slot waveguides [6, 7] offer the advantages of a better layer thickness control and allow for much thinner slot layers [8]. At the same time scattering losses are minimized owing to the smooth layer interfaces. Despite first practical attempts to realize horizontal slot waveguides for nonlinear applications [9], no thorough theoretical study on the optimization of the waveguide geometry has so far been reported. One report [10] theoretically evaluated both vertical and horizontal silicon-based slot waveguide structures with respect to their nonlinear behavior. However, no multi parameter optimization with respect to all geometry parameters was performed in order to find the waveguide geometry that provides maximum nonlinear interaction. Moreover, the low index contrast approximation was used for calculating the effective area instead of the precise equation [4], which can give quantitatively misleading results, as will be discussed in section 3.

In this work, we present a systematic study on the nonlinearity of horizontal slot waveguides operated at a wavelength of 1.55 μm and compare the results with that of vertical slot waveguide structures.

2. Simulation model and technique

Figure 1 shows the cross section of the investigated vertical and horizontal slot waveguide structures. In the case of the vertical slot waveguide we assume for practical reasons that the whole structure is covered with the nonlinear material. Horizontal slot waveguides, on the other hand, can be implemented either with a thin low index layer with the thickness s and a linear refractive index of n _slot embedded between two silicon wires. The lower cladding is a silicon dioxide layer, the upper cladding is either air (n _c1=1) or the same nonlinear material as in the slot region (n _c1=n _slot). In our model, we assume that the nonlinear interaction only occurs in the low index material. The amount of third order nonlinear interaction in a waveguide is expressed by the nonlinearity coefficient γ = k ₀ n ₂/A _eff, where k ₀ is the angular wavenumber, n ₂ the nonlinear refractive index of the nonlinear material, and A _eff the effective area. The effective area can be seen as a figure of merit of how well the waveguide geometry supports the nonlinear interaction. The smaller the effective area provided by the waveguide structure the higher the nonlinear interaction. In high index contrast waveguide structures the effective area is defined by [4]

 

Fig. 1. Cross sections of the three investigated implementations of slot waveguides filled and/or covered with a low index optically nonlinear material: (a) vertical slot waveguide, (b) and (c) horizontal slot waveguides.

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where Z ₀ is the free space wave impedance, n_NL the refractive index of the nonlinear material, ℰ⃗(x,y) the vectorial electric, and ℋ⃗(x,y) the vectorial magnetic field profiles of the waveguide mode of interest. In the context of nonlinear interaction the modes of interest are the TE-like mode for the vertical slot waveguide and the TM-like mode for the horizontal slot waveguide. The upper integral extends over the whole cross section D _total, whereas the lower integral is limited to the area covered by the nonlinear material D _NL.

We employed a commercial full-vectorial 2D finite element method (FEM) based mode solver [11] for the calculation of the waveguide eigenmodes. The 4×4 μm simulation domain was surrounded by 0.4 μm thick perfectly matched layers. An adaptive mesh refinement was used to ensure sufficient accuracy. In order to find the waveguide cross section that provides a minimum effective area, i.e., maximum nonlinear interaction, for a given slot thickness we implemented an iterative search. In a first step, a course grid of width and height values was defined. Then, the corresponding effective areas were calculated using the FEM tool. Next, a 2D fit was performed followed by a search for the minimum effective area. Around this minimum, the grid of width and height values was refined and the effective areas were calculated again. This procedure was repeated until the difference of the obtained minimum effective areas between two iterations was smaller than 10^-4 μm² and until the resolution of the grid was better than 1 nm.

3. Results

First, we studied a horizontal slot waveguide slab system as depicted in the inset of Fig.2(a), where the slot is filled with an optically nonlinear material. The refractive index of the nonlinear material is set to 1.46. By solving the analytic eigenmode equation of this five layer system [12] the electric and magnetic mode field profiles of the TM mode in y-direction can be calculated. Figure 2(a) plots the optical power confined in the slot region and the effective area per length unit in x-direction as a function of the silicon layer thickness h for different slot thicknesses s ranging from 10 to 80 nm. A silicon layer thickness of 160 nm provides maximum confinement and, thus, maximum nonlinear interaction irrespective of the slot thickness. Figure 2(b) shows the power confinement in the slot region and the effective area as a function of the slot thickness at a constant silicon layer thickness of 160 nm. Maximum nonlinear interaction is obtained for slot thickness of s = 15 nm. The dashed lines in Fig. 2(b) and (c) show the low index approximation used in [10] demonstrating the significant deviation of the absolute value of A _eff and the location of the minimum with respect to the height h and the slot thickness s.

Next, we studied the nonlinear behavior of vertical slot waveguide structures (see Fig. 1(a)) with respect to the strongly confined TE-like mode for different refractive indices of the nonlinear cover medium ranging from 1.5 to 1.8. Using the optimization procedure described above, we determined the minimum achievable effective area as a function of the slot thickness s (see Fig. 3(a)). Figure 3(d) plots the corresponding values of the optimized geometry parameters h and w. The results for s > 50 nm match well with those reported by Koos et al.[4], who restricted their simulations to this practically relevant area. For the sake of comparison with the horizontal slot waveguide structure, we considered also smaller slot widths with s < 50 nm. The smallest effective areas are achieved at slot thicknesses around 20 nm, which is in good agreement with the result obtained for the slab system. The corresponding optimum geometry parameters are around 270 nm in vertical and 190 nm in lateral direction.

 

Fig. 2. (a) Cross section of a horizontal slot waveguide slab system and intensity profile of the TM mode; (b) optical power in the optically nonlinear slot region and effective area as a function of the silicon layer thickness for different slot thicknesses, and (c) as a function of the slot thickness at a constant silicon layer thickness of 160 nm. The dashed lines in (b) and (c) represent the low index approximation as used in [10].

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Figures 3(b) and (e) show the minimum achievable effective areas and the corresponding optimum geometry parameters for the strongly confined TM-like modes in horizontal waveguide structures filled and covered with the nonlinear material (see Fig. 1(b)). The results are almost identical to that of vertical slot waveguides both in terms of minimum achievable effective areas and in terms of optimum geometry parameters. This is due to the fact that the refractive index profiles of the cladding surrounding the two silicon wires are very similar for these two configurations. For a nonlinear material with a refractive index of 1.46 the results would be identical. If we do not cover the horizontal slot waveguide structure with the nonlinear material and use air as cladding instead, the confinement of the light in the slot is improved. As the results on the minimum achievable effective area as a function of the slot thickness s plotted in Fig. 3(c) reveal, the additional nonlinear interaction due to the stronger confinement in the slot region outbalances the nonlinear interaction in the cladding region in the case of a structure covered with the nonlinear material. The higher refractive index of the nonlinear material, the stronger this effect becomes. The values of the optimized geometry parameters are plotted in Fig. 3(f). The smallest effective areas range from 0.027 μm² to 0.05 μm² depending on the slot index and are obtained for slot thicknesses of around 15 to 25 nm with silicon wires of ~210 nm height and ~220 nm width.

Finally, we wanted to clarify how the width of an air-covered horizontal slot waveguide influences the nonlinear interaction. In order to answer this question, we used our optimization procedure with respect to the waveguide height h and slot thickness s to determine the minimum achievable effective area as a function of w.

 

Fig. 3. Dependence of the minimum achievable effective area on the slot thickness s for different refractive indices of the nonlinear material in (a) a vertical slot waveguide completely covered with the nonlinear material, (b) a horizontal slot waveguide filled and covered with the nonlinear material, and (c) a horizontal slot waveguide filled with the nonlinear material and covered with air. At each point, the geometry is optimized with respect to the parameters h and w such that a minimum effective area is obtained. The corresponding values of the optimized geometry parameters are plotted in (d)-(f). The results in (a) and (d) for slot thicknesses s > 50 nm match with those published in [4].

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Fig. 4. (a) Dependence of the minimum achievable effective area in a horizontal slot waveguide on the width w for different refractive indices of the nonlinear material filled in the slot region. The upper cladding is air. At each point, the geometry is optimized with respect to the waveguide height h and slot thickness s. The corresponding values of the optimized geometry parameters are plotted in (b).

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The results plotted in Fig. 4(a) indicate that the waveguide width can be considerably increased to values of 400 to 500 nm without significantly sacrificing nonlinearity. This can help reducing scattering losses induced by rough waveguide side walls. The optimum slot thickness remains almost constant over the plotted range of w, while the optimum height decreases rapidly to a value of 160 nm (see Fig. 4(b)).

4. Conclusion

The presented eigenmode analysis of vertical and horizontal slot waveguide structures provides the optimized set of geometry parameters for which a minimum effective area and, thus, maximum nonlinear interaction is achieved. Our results indicate that air-covered horizontal slot waveguides facilitate the exploitation of nonlinear effects in nonlinear materials with a comparatively low refractive index because they allow for much thinner slot layers and thus smaller effective areas than vertical slot waveguides. Ultra-thin vertical slots not only represent a challenge in fabrication but would also be difficult to fill with nonlinear materials. In addition, air-covered vertical slot waveguides appear difficult to realize from a technological point of view, thus prohibiting higher light confinement and smaller effective areas. Horizontally sandwiched structures with very thin slot layers, on the other hand, can be fabricated by CVD methods [9, 13, 14], which are also applicable to organic nonlinear materials [5]. Incorporating for example the highly nonlinear organic material PTS, which has a linear refractive index of n _slot=1.7 and a nonlinear refractive index of n ₂=2.2×10^-16 m²W^-1 [4], in an optimized horizontal slot waveguide structure will give a nonlinearity coefficient γ=n ₂ k ₀/A _eff of >2×10⁷ W^-1km^-1, where we assumed an A _eff of 0.04 μm² based on the results in Fig. 3(c).