1. Introduction

High-index contrast, high nonlinearity silicon-on-insulator (SOI) waveguides have been intensely studied for the past decade, due to their promising nonlinear photonics applications [1]. Much progress has been made using the nonlinearity of silicon waveguides in areas such as all-optical signal processing [2–4] and more recently, quantum photonics [5–7]. However, the promise of nonlinear photonics using silicon based waveguides is often tempered by the poor bulk material TPA figure-of-merit ${\text{FOM}}_{\text{TPA}}^{\text{Bulk}}=\frac{n_2}{{\alpha }_2\lambda }$ which is about 0.4 at 1550 nm. Intensity dependent two-photon absorption places limits on the observations of nonlinear optical processes in silicon, since such processes often require high intensities. Several directions are currently being pursued in order to mitigate this problem. Some examples include moving the operating wavelength beyond the TPA limit [8], limiting the TPA induced free-carrier absorption [9] and using other highly nonlinear silicon based materials with limited TPA [10,11] (e.g. hydrogenated a-Si, silicon nitride). Silicon-organic hybrid slot waveguides offer a promising alternative by confining the power of the fundamental TE mode in the slot between the two silicon cores, thereby circumventing the low ${\text{FOM}}_{\text{TPA}}^{\text{Bulk}}$ of silicon. Nonlinear optical functionality is obtained by filling the slot with highly nonlinear organic polymers with negligible TPA. Experimental and numerical studies have shown that such waveguides can have waveguide TPA figure-of-merit FOM_TPA of up to 2.19 and 4.25 respectively [12, 13]. However, the overall efficiency of nonlinear processes such as FWM is dependent on the effective interaction length, which is a function of the linear and nonlinear losses. Asymmetric slot waveguides fabricated in [14] showed relatively low loss relative to symmetric slot waveguides, with some sacrifice of the waveguide nonlinearity. A systematic numerical study on the trade-off between roughness scattering loss and TPA figure-of-merit has yet to be done.

In this article, we study the effects of the slot waveguide geometry on FOM_TPA and the oughness scattering loss α. We vary the waveguide and slot widths and allow for asymmetrical slots, so as to determine the optimal silicon-organic hybrid slot waveguide geometry. The waveguide height h is chosen to be 220 nm for compatibility with most common rectangular strip waveguide heights [15–17]. The effect of waveguide height on the optimal geometry is discussed further in the Appendix. In all the calculations and discussion, we use the fundamental TE mode of the slot waveguide. We note that although the results obtained are with respect to χ⁽³⁾ processes, they are nonetheless of relevance to nonlinear slot waveguides in general (e.g. slot waveguide amplifiers [18] and slot waveguide modulators [19, 20]).

2. Slot waveguide geometry and TPA figure-of-merit

SOI slot waveguides typically consist of two closely spaced silicon “cores” separated by a narrow slot, on top of a SiO₂ buried oxide layer (see Fig. 1). For applications in nonlinear photonics, a highly nonlinear polymer fills the slot and acts as the waveguide cladding. The type of nonlinear polymer is chosen to have a large nonlinear refractive index n₂ and a small TPA coefficient α₂. The waveguide nonlinear parameter γ_wg, which is proportional to the nonlinear refractive index, quantifies the level of χ⁽³⁾ processes in the waveguide. Slot waveguides generally have large γ_wg, due to the high confinement of the mode within the slot. The quantity of interest is the waveguide TPA figure-of-merit ${\text{FOM}}_{\text{TPA}}=\frac{-\mathrm{Re}({\gamma }_{wg})}{4\pi \mathrm{Im}({\gamma }_{wg})}$, where γ_wg is given by [21]

 

Fig. 1 Silicon-organic hybrid slot waveguide geometry. The nonlinear polymer cladding is DDMEBT, the waveguide height h is 220nm.

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Here, Z₀ is the free-space impedance, $\overrightarrow{E}$ and $\overrightarrow{H}$ are the modal electric and magnetic fields, ω is the frequency of operation, c is the speed of light. The third-order nonlinear susceptibility χ⁽³⁾ is a function of the co-ordinates x, y due to the different nonlinearities of the cores and cladding. We study the effect of the geometry on FOM_TPA by varying the slot width and both core widths independently, thus taking into consideration asymmetric slots. The nonlinear optical polymer chosen for the calculations is DDMEBT, with the material properties at 1550nm as follows: linear and nonlinear refractive indices n₀ = 1.8 and n₂ = 2.0 × 10⁻¹⁷ m²W⁻¹, ${\text{FOM}}_{\text{TPA}}^{\text{Bulk}}=5$ [12,13,22]. The silicon material properties are taken to be: n₀ = 3.47, n₂ = 5.0×10⁻¹⁸ m²W⁻¹, and α₂ = 8×10⁻¹² m/W, giving ${\text{FOM}}_{\text{TPA}}^{\text{Bulk}}=0.4$ [12]. The SiO₂ refractive index is n₀ = 1.44. The relation between the nonlinear susceptibility, and the nonlinear refractive index and TPA coefficient is

Figure 2 shows contour plots of the calculated FOM_TPA when slot width and rail widths (w₁ and w₂) are varied. The waveguide field profiles are calculated using a commercial mode solver. Redundant data due to the symmetry of the slot waveguide are excluded. Figure 3 shows contour plots of the real part of the waveguide nonlinear coefficient Re(γ_wg) [W⁻¹ m⁻¹], as given by Eq. (1), so as to give a complete picture of the slot waveguide nonlinearity. A blue background behind the contours indicates single-mode geometries and an amber background indicates multi-mode geometries. As can be seen from the contours, smaller waveguide cores generally have higher FOM_TPA values. On the other hand, changing the slot size has a smaller effect on the figure-of-merit, with smaller slots having marginally higher FOM_TPA. As seen from the integral in Eq. (1), the nonlinear parameter can be separated into contributions from the core regions and the cladding, that is Im(γ_wg) = Im(γ_core) + Im(γ_clad) [13]. Naturally, a larger silicon cross-section should result in a poorer figure-of-merit, which is in line with our numerical result. At the same time, smaller core sizes also avoid multi-moded regimes which may perform poorly due to inter-modal coupling, modal dispersion and poor mode overlap between fundamental and higher order modes. However, slot waveguide geometry also has a strong effect on roughness scattering loss, which the above analysis does not take into account. In the next section, we see that certain geometries are more adversely affected by sidewall roughness.

 

Fig. 2 Contour plots of FOM_TPA with varying slot waveguide geometries. Data above the anti-diagonals are not plotted due to the symmetry of the structure. A blue background indicates single-mode geometries and an amber background indicates multi-mode geometries.

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Fig. 3 Contour plots of Re(γ_wg) [W⁻¹ m⁻¹] with varying slot waveguide geometries. Data above the anti-diagonals are not plotted due to the symmetry of the structure. A blue background indicates single-mode geometries and an amber background indicates multi-mode geometries.

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3. Slot waveguide geometry and roughness scattering loss

Sidewall roughness is a geometrical imperfection inherent to the fabrication of sub-micron silicon waveguides. The sidewall surface roughness acts as a perturbation that couples the forward propagating waveguide modes to backward modes and free-space modes. Modern fabrication techniques can reduce roughness scattering loss of silicon wire waveguides to close to 0.5dB/cm [16, 17]. In contrast, slot waveguides generally show losses on the order of 5dB/cm [23], with 2dB/cm loss reported for certain geometries with asymmetric or wider cores [14, 16]. Since the efficiency of nonlinear optical processes are highly dependent on the effective interaction length, which is a function of the loss, it is desirable to find a balance between roughness induced losses and waveguide nonlinearity.

A number of analytical and numerical methods have been proposed to study the effect of sidewall roughness on waveguide propagation loss [24–26]. To study the influence of slot waveguide geometry on roughness scattering loss, we used a numerical roughness loss estimation method from [27], implemented using a commercial FEM software. We chose this method due to its ease of implementation, accuracy and generality. The input parameters for the scattering loss calculation are the modal fields and effective indices at 1550nm, as well as experimentally extracted roughness data. Taking into account published experimental measurements as well as recent advances in fabrication, we used a correlation length L_c of 50nm and RMS roughness σ of 1nm [15–17, 28]. We also note that the roughness scattering loss is proportional to the square of the roughness i.e. α ∝ σ². This means that the calculated loss can be easily scaled to reflect different values of RMS roughness.

Figure 4 shows the calculated roughness scattering loss in dB/cm for the same geometries used in Fig. 2 and 3. As can be seen, waveguide geometry has a very strong effect on the scattering loss, with small slots generally having larger loss values. Also, the maxima for loss corresponds quite closely to the maxima of the waveguide nonlinearity Re(γ_wg). This can be understood as being a result of the fact that both quantities scale accordingly to the strength of the field within the slot. Given these three quantities, γ_wg, FOM_TPA and α, we are able to calculate the FWM conversion efficiencies for each of the geometries and hence find the optimal cross-section for a nonlinear silicon-organic hybrid slot waveguide.

 

Fig. 4 Plots of roughness scattering loss in dB/cm with varying slot waveguide geometries. Data above the anti-diagonals are not plotted due to the symmetry of the structure.

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4. Discussion

Combining the results from previous sections, it is clear that TPA figure-of-merit and roughness scattering losses both have to be accounted for when choosing a slot waveguide geometry for nonlinear photonics applications. The FOM_TPA should be maximized, while maintaining an acceptable level of scattering loss. More concretely, we can decide on an optimal slot waveguide geometry for FWM applications by calculating the FWM conversion efficiency η for each corresponding cross-section.

To calculate the FWM conversion efficiency, we used an analytical expression that takes into account both linear scattering losses and TPA absorption losses [29, 30].

Here, γ ≡ Re(γ_wg), γ′ ≡ Im(γ_wg), L is the waveguide length, $L^{\prime }=\frac{1-e^{-\alpha L}}{\alpha }$ is the linear loss limited effective length, P₀ is the power in the waveguide at the input, $\overline{P}$ is the path averaged power. For simplicity, we have assumed zero phase mismatch. This is justified since we can choose the wavelength separation between pump and signal to be small enough such that dispersion effects can be neglected. At this point, the waveguide length L is a free parameter that has to be optimized to give the maximum η. To do so, we rewrite Eq. (3) in a less familiar form,

The first term in the equation is 2π times the TPA figure-of-merit, 2πFOM_TPA, which sets the upper limit on η under the assumptions of Eq. (3). We identify two regimes of behaviour with respect to the size of the factor $\frac{\alpha }{{\gamma }^{\prime }{P}_0}$. In the linear loss limited regime, i.e. $\frac{\alpha }{{\gamma }^{\prime }{P}_0}\gg 1$, the factor in the square brackets of Eq. (4) can be expanded to first order to give the familiar result η = (γP₀L′)²e^−αL. The optimum waveguide length in this case is L_opt = 1/α. In the TPA limited regime, i.e. $\frac{\alpha }{{\gamma }^{\prime }{P}_0}\ll 1$, we have to maximize the factor in the square brackets, which has the form ${\left[\frac{\log (1+x)}{1+x}\right]}^2$. This has a maximum value of e⁻² when 1+x = e. Solving for L gives the optimum waveguide length

Note that Eq. (5) is only valid when $0<\frac{(e-1)\alpha }{2{\gamma }^{\prime }{P}_0}\ll 1$. Due to the relatively high roughness scattering loss and high TPA figure-of-merit, the slot waveguides studied are all in the linear loss limited regime and hence we choose L_opt = 1/α. For example, for a comparatively low loss and low FOM_TPA 320nm/180nm (left/right core width) waveguide with 180nm slot, the factor $\frac{\alpha }{{\gamma }^{\prime }{P}_0}=\frac{2.1}{P_0}$. Even in this case, the input power P₀ has to be ≫ 1W for the waveguide to be in the TPA limited regime.

Figure 5 shows the FWM conversion efficiency calculation results with P₀ = 200mW. The maximum conversion efficiency occurs at symmetric waveguide core widths of 220nm with slot widths of 120nm. Too small slots apparently have too great roughness scattering loss, while too large slots have too low waveguide nonlinearity. Asymmetric slot waveguides, despite having lower roughness loss α values in general, are not particularly better suited for nonlinear wave mixing applications due to the corresponding reduction in γ_wg. Interestingly, the overall difference in η considering all geometries is only about 3dB, despite very different α and γ_wg. This is likely a result of having defined the waveguide length as L = 1/α, which is close to the optimum waveguide length in the presence of linear loss. The waveguide lengths used in the calculation ranged from 8mm to 2.4cm. The optimum waveguide cross-section may change if the roughness scattering loss is low enough such that the waveguides are in the TPA limited regime. For a fixed pump power of P₀ = 200mW and considering the 320nm/180nm (left/right core width) waveguide with 180nm slot, the condition is α < 0.18dB/cm. At this very low linear loss level, the optimum waveguide geometry would favor high FOM_TPA.

 

Fig. 5 Plots of FWM conversion efficiency η in dB with varying slot waveguide geometries. Waveguide nonlinearity γ_wg and roughness scattering loss α correspond to the values in Fig. 2, 3 and 4. Pump power is fixed at P₀ = 200mW. Waveguide length is chosen to be the linear loss optimized length L = 1/α.

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Practically, it is of interest to minimize the total length of the waveguide to reduce its on-chip footprint. As such, we re-calculate the FWM conversion efficiencies using a fixed waveguide length of 8mm (i.e. the shortest waveguide length in Fig. 5). Figure 6 shows the calculated η in this case. The optimum geometry is still 220nm symmetric cores, but with a smaller slot width of 100nm. This is because short lengths favor waveguides with higher nonlinearity as there is less distance to accumulate the converted power. Since it is experimentally challenging to fabricate waveguides with precise dimensions, it is also important to understand the sensitivity of the FWM conversion efficiency to waveguide geometry. From Fig. 6, we can see that the conversion efficiency degrades only by a maximum of 0.5dB when the core widths and slot widths vary by ±20nm.

 

Fig. 6 Plots of FWM conversion efficiency η in dB with varying slot waveguide geometries. Waveguide length is L = 8mm.

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From Fig. 5, the FWM conversion efficiencies are already comparable to state-of-the-art rib waveguides [9]. However, we have not fully utilized the high TPA figure-of-merit of these slot waveguides, which would require a higher pump power P₀. To better represent the performance limits of the slot waveguides, we re-calculated the FWM conversion efficiency η versus pump power using experimentally measured loss figures from [16]. Figure 7(a) shows the calculated η in dB for the slot waveguide, as compared to a rib waveguide from [9]. The slot waveguide cross-section in the reference is 260×220 nm (width × height) for both cores, with a slot width of 100 nm, and a measured loss of 2dB/cm. The calculated γ_wg = 437W⁻¹m⁻¹ and FOM_TPA = 2.46. For the rib waveguide, the parameters used are: 650×220nm and 70nm slab, α_dB = 0.68dB/cm, γ_wg = 163W⁻¹m⁻¹ and FOM_TPA = 0.4. We used Eq. (3) and hence have assumed, for simplicity, no free-carrier absorption (FCA) loss. The dotted lines represent the calculated η for a slot waveguide loss of 2 ± 0.5dB/cm. Unsurprisingly, the slot waveguide outperforms the rib waveguide, especially at high pump powers, due to the high TPA figure-of-merit of the former. Note that we used the linear loss optimized length L = 1/α, with the rib waveguide length at 6.4 cm and the slot waveguide length at 2.2 cm. At pump powers above 200 mW, the rib waveguide enters the TPA limited regime and hence we also plot η for the TPA optimized length $L=\frac{e-1}{2{\gamma }^{\prime }}=2.7\text{cm}$.

 

Fig. 7 FWM conversion efficiency η(dB) versus pump power P. Rib waveguide dimensions are 650 × 220 nm with a slab height of 70 nm. Slot waveguide dimensions are 260 × 220 nm for both cores, with slot width of 100nm. (a)η assuming no FCA. (b)η including the effects of FCA with varying free carrier lifetimes.

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The caveat for this result is that the calculated η will not be accurate at very high pump powers approaching 1W in the waveguide, due to the presence of FCA in reality. Such high powers may be achievable using intensity enhancing structures such as micro-ring resonators or photonic crystal waveguides [31, 32]. In Fig. 7(b), we re-calculated the conversion efficiency at high pump powers using coupled amplitude equations that account for the effects of FCA [33]. For sub-micron silicon waveguides, the free carrier lifetime τ ≈ 1ns and hence for the slot waveguide we choose τ = 1 ± 0.5ns. For reverse biased p-i-n junction rib waveguides, the free carrier lifetime is significantly reduced. Based on simulation results in [9], we choose τ to range from 20ps to 100ps. As can be seen, the slot waveguide still performs favorably compared to the rib waveguide under these realistic simulation conditions. Moreover, there exists the possibility of incorporating a thin silicon slab with the slot waveguide to allow removal of free carriers.

At such high pump powers, there is also a risk of thermal damage to the nonlinear optical polymer due to the localized heating within the slot. In [22], thermal damage is observed at temperatures in excess of 125°C. To provide an understanding of the heating effect within the slot, we numerically solved the 2D steady state heat equation assuming the total heating power to be equal to the loss due to TPA [34]. The TPA loss at the start of the waveguide is given by 2γ′P₀. For the slot waveguide from Fig. 6, the TPA loss for P₀ = 1W is 1.27dB/cm which is 246 mW over 1 cm. This value was used to calculate the heat source power density, which we assumed to be uniformly distributed over the two silicon waveguide cores. The thermal conductivities used were, k_Si = 149W/(m.K), $k_{Si{O}_2}=1.4\mathrm{W}/(\mathrm{m}.\mathrm{K})$ and k_poly = 0.1W/(m.K). Figure 8 shows the calculated temperature change, which is 14.6K within the slot. This unsophisticated numerical estimate indicates that within the range of pump powers considered here, the temperature rise due to TPA heating is insufficient to damage the optical polymer under normal operating conditions.

 

Fig. 8 Steady state temperature change ΔT in the silicon slot waveguide under TPA heating at 1W pump power. The heat source has a uniform density of 0.215mW/(µm³) within the silicon cores.

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5. Conclusions

In conclusion, we have studied the effects of silicon-organic hybrid slot waveguide geometry on the FOM_TPA and roughness scattering loss α. We have combined these results and determined the optimum slot waveguide cross-section for maximum FWM conversion efficiency η, which was found to have symmetrical 220nm waveguide cores with 120nm slots. We calculated that the conversion efficiency degrades only by 0.5dB when the core widths and slot widths vary by ±20nm. We also showed that the performance of the slot waveguide was better as compared to state-of-the-art rib waveguides, especially at high pump powers where TPA effects become important. This remains true even when accounting for the effects of FCA. Finally, we determined that the heating in the slot waveguide due to TPA was insufficient to damage the nonlinear optical polymer for the range of powers considered.

The results and methods from this study are specific to FWM applications, but are generalizable to other nonlinear optical effects where roughness loss and TPA loss are important. We also note that even better conversion efficiencies could be obtained by using nonlinear polymers (e.g. PTS) which display anomalous refractive index dispersion [35]. However, some authors have described the difficulty in fabricating thin films of PTS [36] as compared to DDMEBT.

Appendix: Effect of waveguide height on optimal geometry

To study how waveguide height affects optimal geometry, we calculated the FWM conversion efficiency for two additional waveguide heights of 180nm and 260nm using Eq. 3. The calculation results are shown in Fig. 9. We incorporated the corresponding γ_wg, FOM_TPA and α of the new waveguide geometries. In general, smaller height h leads to lower α and hence higher η. For h = 180nm, the optimum geometry occurs at symmetric waveguide core widths of 240nm with slot widths of 120nm. For h = 260nm, the optimum geometry occurs at symmetric waveguide core widths of 200nm with slot widths of 120nm. This means that the optimum geometry does shift when h is changed. Nevertheless, a waveguide height of 220nm remains a prudent choice to facilitate mode matching with most common rectangular strip waveguides, as mentioned previously. Moreover, 220nm height strikes a balance between providing enough mode confinement, thus limiting substrate leakage and bending loss, while not being too thick such as to increase surface roughness scattering. From the point of view of fabrication tolerance, the optimum geometry as suggested in the main body of the text is relatively robust to changes of height h. This can be seen by the fact that height changes of Δh = ±40nm only marginally shift the optimum point.

 

Fig. 9 Plots of FWM conversion efficiency η in dB with varying waveguide heights. Pump power is fixed at P₀ = 200mW. Waveguide length is chosen to be the linear loss optimized length L = 1/α.

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Acknowledgments

The authors acknowledge the support of the National Research Foundation Singapore under its Competitive Research Programme (Grant No. NRF-CRP 14-2014-04). We would like to thank Chu Hong Son and Jason C.E. Png for helpful discussions. We also thank the reviewers for insightful comments and suggestions.

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