1. Introduction

The slot waveguide [1, 2] is a promising structure for the realization of active silicon photonic devices. In a slot waveguide structure, guided light with the major electric field component perpendicular to the slot interface can be strongly confined in a narrow low-index gap between two high-index regions. This strong light confinement in the slot enables the optimization of existing and the realization of novel silicon photonic device concepts [3, 4, 5, 6, 7, 8, 9, 10, 11].

Vertical slot waveguides with narrowbending radii can be fabricated from a single monocrystalline silicon layer [12] but put high demands on the fabrication process due to small feature sizes. Moreover, side wall roughness is a critical issue. On the other hand, horizontal slot waveguides [13] offer the advantages of a better layer thickness control and large waveguide width-to-height ratios. At the same time scattering losses are minimized owing to the inherently smooth layer interfaces. Horizontal rib-type slot waveguides [14] (see Fig. 1) are of particular interest because they facilitate electrical wiring, which is a prerequisite for active photonic devices. However, these waveguides can suffer from leakage of the TM-polarized slot mode into the outer slot waveguide slab due to TM-TE mode coupling, which renders them practically useless. In this study, we investigate the lateral leakage mechanism in symmetric rib-type SOI slot waveguides at a wavelength of 1.55 µm employing semi-analytical and numerical methods.

2. The VMM method

The variational mode-matching (VMM) method used in the computation of the optical modes in the waveguide cross-section belongs to the family of fully vectorial film mode-matching methods [15, 16] where the optical field is expanded in terms of local TE-and TM-polarized film modes. One of the most critical steps in mode-matching techniques is the determination of a proper set of local eigenmodes—especially when perfectly-matched layers (PMLs) acting in the thickness dimension are present—since an improper truncation of the spectrum of local waveguide modes causes numerical instabilities. In the VMM method these difficulties are overcome by employing a Galerkin scheme to the local 1D waveguide problem which reduces the computation of the relevant local waveguide modes to the solution of a well-understood linear eigenvalue problem [17].

 

Fig. 1. (a) Cross section of a rib-type slot waveguide, (b) overall optical power (contour) and electric field (arrows) distribution of the TM-like slot mode, (c) minor (TE) components.

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Compared to other simulation techniques like finite-elements or finite-differences, the VMM method appeals much more to device physics. Therefore, the computational cost is moderate since the number of unknowns can be kept rather low. However, the more striking advantage provided by the VMM method arises from the fact that the simulation of lateral leakage can be performed without PML layers acting in the lateral direction. Due to the nature of the modal expansion, the physically correct radiation boundary conditions are inherent to the VMM model.

3. TM-TE mode coupling

Leakage due to TM-TE coupling was first described for various waveguide structures in the microwave wavelength range [18]. Recently, this important issue has been brought up again in the investigation of SOI rib waveguides [19], where the practical relevance of waveguiding structures with sub-micron feature-size was shown. Lateral leakage of a TM-like rib waveguide mode occurs if its effective index is smaller than that of a TE slab mode outside the rib. In this case, the minor TE field components of the TM-like rib mode excite the TE slab mode propagating away from the waveguide core. In the case of the fundamental TE-like rib mode this effect does not occur because its effective index is always larger than that of TM-slab modes outside the rib.

In rib-type slot waveguides the situation is more complex. Slot waveguides represent a strongly coupled system consisting of two high-index waveguide cores resulting in two firstorder modes for each polarization [20]: an even mode with the major electric field components in the upper and lower waveguide layer being parallel, and an odd mode with the major electric field components in the upper and lower waveguide layer being anti-parallel. Depending on the geometry parameters and the wavelength only the even mode exists. For the TM polarization the even mode corresponds to the highly confined slot mode. Contrary to the major electric field components, the minor electric field components of the TM-like slot mode are anti-parallel (see Fig. 1 (b) and (c)). The role of the two TE-slab modes in the TM-TE leakage mechanism has to be studied against this background.

Figure 2(a) and (b) show the effective indices of the TM-like slot mode and the TE-slab modes as a function of h/H for structures with H=137 nm, w=400 nm, and slot thicknesses S=150 nm and S=50 nm, respectively. The thickness H corresponds to the value at which the fraction of optical power confined in the slot of a slot waveguide slab system is maximum.

 

Fig. 2. Effective indices of the TM-like slot mode, and the even and odd TE slab modes as a function of the relative slab thickness h/H for H=137 nm, w=400 nm and for two slot thicknesses, (a) S=150 nm and (b) S=50 nm at λ=1.55 µm; (c) and (d) show the lateral leakage losses of geometries with corresponding w and S, and the leakage criteria obtained numerically from the real part of the effective indices of the VMM simulation (upwardpointing triangles) and using the effective index method (downward-pointing triangles). The vertical dashed lines indicate the thickness used for the computation of the effective indices in (a) and (b).

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The effective index N _eff,TM of the TM-like slot mode was calculated numerically using the variational mode-matching (VMM) method, whereas the effective indices n ^(slab) _eff,TE,even, n ^(slab) _eff,TE,odd of the even and odd TE slab modes were determined semi-analytically by solving the transcendent eigenmode equation of a five layer slab system. Three regions for h/H can be distinguished:

In region I, coupling of the TM-like slot mode to both TE slab modes is possible, in principle, whereas in region II only the even TE slab mode can be excited. In region III, no TM-TE coupling can occur. Figure 2(c) and (d) plot the leakage values derived from the imaginary part of the effective index of the TM-like slot mode calculated by the VMM simulations. The vertical dashed lines indicate the thickness used for the computation of the effective indices in Fig. 2(a) and (b). A comparison of the leakage values along the line in Fig. 2(c) with the refractive index values shown in Fig. 2(a) reveals that the abrupt change of the leakage loss matches with the boundary between region I and II at h/H=0.42, where

For structures with smaller values of h/H the lateral leakage loss becomes zero, although the effective index of the even TE slab mode still exceeds that of the TM-like slot mode. The boundary between region II and III is defined by

Due to the perfect symmetry of the slot waveguide structure in vertical direction, the minor field of the even fundamentalTM mode has no even components which could couple to the even TE-slab mode. As shown in Fig. 2(b), for a slot thickness of S=50 nm the effective index of the odd TE-slab mode lies below the effective index of the TM-like slot mode over the whole range of h/H and consequently no leakage occurs (see Fig. 2(d) region II). Calculating the relative thickness h/H where the relationship (1) is fulfilled at other waveguide thicknesses H provides a leakage criterion for perfectly symmetric rib-type slot waveguides, which agrees well with the abrupt changes of the leakage losses in Fig. 2(c) and (d).

4. Semi-analytic leakage criterion

Lateral leakage in a perfect symmetric rib-type slot waveguide only occurs above a critical relative slab thickness h/H for which equation (1) is fulfilled. Latter can be calculated semi-analytically by solving the transcendent eigenmode equation of a five layer slot waveguide slab, whereas the effective index of the TM-like slot mode has to be determined by fully vectorial 2D mode solvers. In order to avoid a time-consuming analysis we approximate the effective index of the TM-like slot mode by solving the transcendent eigenmode equation of a slot waveguide slab system for the inner and outer region of the rib-type slot waveguide structure and applying the well-known effective index method [21]. In Fig. 2(c) and (d) the leakage criterion using the real part of the effective index of the VMM simulation (upward-pointing triangles) is compared with the leakage loss values obtained from the same simulations. This numerical criterion agrees well with the abrupt change in leakage loss values, demonstrating the validity of the stated criterion. Moreover, the semi-analytic criterion (downward-pointing triangles) is almost identical with the numerical findings obtained by the VMM method even for the rather small waveguide width of w=400 nm. Therefore, a computationally inexpensive and, nevertheless, sufficiently accurate elaboration of design rules for leak-proof rib-type slot waveguides is facilitated by semi-analytical methods.

 

Fig. 3. Leakage loss of rib-type slot waveguides with thickness variations of ±5 nm in the upper rib structure for (a) S=150 nm and (b) S=50 nm at λ=1.55 µm. The semi-analytical leakage criteria for odd (upward-pointing triangles) and even (downward-pointing triangles) TE slab modes are calculated for the most critical structure H′=H-5 nm and h′=h+5 nm. The crosses represent the leakage criteria of a perfectly symmetric system.

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5. Influence of asymmetries

In practice, perfectly symmetric structures are not achievable and variations of the geometry parameters caused by fabrication processes have to be taken into account. These variations lead to an even contribution in the minor field of the TM-like slot mode, thus enabling the coupling of the TM-like slot mode with the even TE slab mode. In order to study the influence of the deviations on the leakage loss the thicknesses h′ and H′ of the upper rib waveguide are varied by ±5 nm (see inset of Fig. 3(a)) resulting in four different geometries. For each geometry the leakage loss is calculated with theVMMsolver and the maximum value is plotted for each point of H and h/H in Fig. 3. Due to the asymmetry induced by the deviations, waveguide geometries in region II become lossy. Although much lower than in region I these leakage losses can limit the practical usability of waveguides in region II. Taking into account the slight asymmetry of the structure, the semi-analytic solutions of the leakage criteria (1) for the odd (upward-pointing triangles) and (2) for the even (downward-pointing triangles) TE modes agree well with the abrupt change of the leakage values calculated by VMM. Both criteria are calculated for the most critical geometry, which is H′=H-5 nm and h′=h+5 nm. With respect to the leakage criteria of the perfectly symmetric system (crosses) only the leakage criterion for the even TE slab mode shows a significant shift to lower values of h/H.

 

Fig. 4. (a) Influence of the slot thickness S at w=700 nm, and (b) of the rib width w at S=50 nm on the the leakage criteria for coupling to the even (dashed lines) and odd (solid lines) TE slab modes.

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6. Influence of geometry

The leakage criteria for coupling to the even and odd TE slab modes depend strongly on the geometry parameters, as shown in Fig. 4. With decreasing slot thickness S the criteria shift to higher values of h/H (see Fig. 4(a)). Moreover, region II expands and region I vanishes over an increasingly large range of H, thus enabling leak-proof perfectly symmetric geometries for arbitrary h/H. The dependence on the waveguide width w is less pronounced because the width has only a minor impact on the effective index of the slot mode. The influence of w on the leakage criteria is inverse to that of the slot thickness. For wider waveguides the leakage criteria shift to higher values of h/H.

7. Resonance effects

It has been demonstrated that resonance effects can reduce the horizontal leakage losses due to TM-TE coupling for certain waveguide widths in otherwise leaky rib waveguide geometries [18, 19]. These resonance effects originate from a destructive interference of TE waves generated by TM-TE mode conversion at the rib walls. The resonant widths can be calculated using the effective indices N _eff,TM of the TM-like rib mode and n ^(Core) _eff,TE of the TE slab mode of the layer system in the core region. In the case of rib-type slot waveguides these resonance effects can also be exploited to suppress lateral leakage due to TM-TE coupling. However, resonances related to the even and odd slab modes occur at different widths:

where m is an integer with m>0.

For a perfectly symmetric waveguide geometry in region I the cancellation of the coupling to the odd TE slab mode at widths given by (4) results in an efficient suppression of lateral leakage. Slight variations of the waveguide geometry give rise to even TE field components and, thus, coupling to the outer slab occurs again, which prevents a complete leakage cancellation. The exploitation of the resonance effect is of particular interest for waveguide geometries with large slot widths having a large region I. As an example, Fig. 5(a) shows the leakage loss of a waveguide geometry with S=100 nm, H=140 nm, and h=105 nm, i.e., h/H=0.75, as a function of the waveguide width. Structural deviations of ±5 nm lead to small shifts of the resonance widths, which are caused by a change of the effective indices. Due to the narrow shaped resonances these shifts translate into a sharp increase of leakage losses.

From a practical point of view, resonance effects are more interesting for waveguide geometries in region II, where leakage losses occur only in the presence of slight asymmetries via coupling to the even TE slab mode. Since these leakage losses are much smaller than in region I (see Fig. 3) shifts of the resonances caused by deviations of the geometry have less impact on the propagation performance. Therefore, waveguide designs with widths fulfilling the resonance condition offer an improved fabrication tolerance because waveguides with deviations from the perfectly symmetric design operate in the vicinity of the resonant minimum, where losses are still small. Figure 5(b) plots the leakage loss of a waveguide geometry with S=50 nm, H=140 nm, and h=84 nm, i.e., h/H=0.6 (indicated by a star in Fig. 4(b)). The period of the widths at which cancellation occurs according to (3) is reduced because of the higher index difference between the even TE slab mode of the core region and the TM-like slot mode compared to the structure in Fig. 5(a).

8. Conclusion

We studied the lateral leakage mechanism in symmetric SOI rib-type slot waveguides, which hold the promise to enable a new category of active silicon photonic devices.We demonstrated that by solving the transcendent eigenmode equation of the slot waveguide slab system and using the effective index method a leakage criterion for symmetric rib-type slot waveguides can be defined. We proved the validity of this approach by comparing with rigorous numerical simulations employing VMM. Taking structural deviations into account a stricter criterion for laterally leak-proof rib-type slot waveguide was obtained. Furthermore, we demonstrated that the parameter region of leaky geometries can be of interest by exploiting resonance effects at certain widths, which reduce losses by orders of magnitude in the otherwise leaky region. This enables the use of shallower structures with larger h/H for applications where ultra-low loss operation is not a prior issue. For example, such structures are advantageous in self-suspended configurations, for sensing or for mechanically tunable devices. Moreover, the requirements towards lithography are relaxed because single mode operation of these shallow structures is guaranteed also for wider waveguides.

 

Fig. 5. Resonant cancellation of the leakage to the odd TE slab mode in region I (a) and the even TE mode in region II (b). The dashed lines indicate the waveguide widths calculated semi-analytically for the perfectly symmetric structures, i.e., H′=H and h′=h, at which resonances occur for odd and even TE modes, respectively.

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