Figure 1A depicts the cross-section view of a typical SOI wafer to form various silicon waveguides (e.g. SWGS waveguides). The SOI wafer considered here has a 340-nm top silicon layer and a 2-μm Silica (SiO_{2}) buried oxide (BOX) layer. Figure 1B depicts the top view structure of the proposed SWGS waveguide. The corresponding 3D structure is illustrated in Figure 1C. SiO_{2} with 1-μm thickness is used to fulfil and cover the silicon-based SWGS waveguide. The SWGS waveguide takes advantage of both slot waveguide and SWG waveguide. The mode guiding mechanism, i.e. SWGS mode, can be regarded as the combination of surface-enhanced supermode and Bloch mode. The high refractive index contrast and resultant electric field discontinuity at the slot boundaries (silicon-SiO_{2}) and the narrow slot region lead to the surface-enhanced supermode. The periodic structure of silicon-SiO_{2} with subwavelength periodicity leads to the Bloch mode. The proposed SWGS waveguide features reduced nonlinearity due to tightly confined SWGS mode in the SiO_{2} slot region and less residual light in the silicon region.

Figure 1: Schematic illustration of silicon platform and SWGS waveguide.

(A) Illustration of an SOI wafer to form various silicon waveguides. (B) Top-view structure of the SWGS waveguide. (C) 3D structure of a SWGS waveguide.

Remarkably, in order to couple light from the strip waveguide into SWGS mode of the SWGS waveguide, a strip-to-SWGS mode converter is designed, which is composed of a strip-to-slot mode converter [13] and two strip-to-SWG mode converters [4], as illustrated in Figure 2. First, the strip-to-slot mode converter is based on a stable Y branch taper structure through mode-matching method. To obtain higher conversion efficiency, we choose the method of adiabatic coupling. Then, the slot mode is converted into the SWGS mode. Two triangle taper structures, each enabling the conversion from the strip mode to the SWG mode (Bloch mode), are employed to implement the evolution from the slot mode to the SWGS mode.

Figure 2: 3D structure of the strip-to-SWGS mode converter, consisting of one strip-to-slot mode converter and two strip-to-SWG mode converters.

We choose the taper length of Y branch taper structure and triangle taper structure large enough to obtain conversion efficiency close to 100%. The SWGS waveguide geometric parameters are set as follows: silicon width: 375 nm, slot width: 100 nm, period: 200 nm, duty cycle: 50%. Figure 3A shows the top view of the simulated electric field distribution. The electromagnetic fields of strip waveguide, SWGS waveguide, and the mode conversion region between them are accurately calculated using the three-dimensional (3D) finite-difference time-domain (FDTD) method. In the simulations, the coupling efficiency is as high as 99.3%, which indicates the mode conversion loss to be only 0.0305 dB. We further monitor the mode evolution process from strip mode to slot mode and to SWGS mode, as shown in Figure 3B–G.

Figure 3: Simulation results for strip-to-SWGS mode converter.

(A) Simulated top view of electric field distribution of the strip-to-SWGS mode converter and (B–E) cross-section view of mode evolution. (B) Strip waveguide region. (C) Middle of the strip-to-slot mode converter. (D) Slot waveguide region. (E) Middle of the slot-to-SWGS mode converter. (F) Si segment of SWGS waveguide. (G) SiO_{2} segment of SWGS waveguide.

The mode properties of conversion process are analyzed also using the 3D FDTD method. We further show in Figure 4A–F the mode normalized intensities along the x and y directions of the guided and propagated fundamental TE mode in six cross-sections corresponding to Figure 3B–G, respectively. It is shown in Figure 3B–G and Figure 4A–F that the slot mode and SWGS mode are confined outside the silicon region. In particular, SWGS mode is further delocalized from the silicon region. We calculate and compare the light concentration ratio of the slot waveguide and Si segment of SWGS waveguide, defined by the ratio of the light confined in the slot region to that of the total light. The light concentration ratio is ~22% for the slot waveguide while ~24.8% for the Si segment of SWGS waveguide. Moreover, for the SiO_{2} segment of SWGS waveguide, almost all the light is within the low refractive index SiO_{2} region. Hence, it is expected that the SWGS waveguide features reduced nonlinearity due to great light delocalization from the silicon region.

Figure 4: Simulated mode normalized intensities along the x and y directions of the guided and propagated fundamental TE mode in strip-to-SWGS mode converter.

(A) Strip waveguide region. (B) Middle of the strip-to-slot mode converter. (C) Slot waveguide region. (D) Middle of the slot-to-SWGS mode converter. (E) Si segment of SWGS waveguide. (F) SiO_{2} segment of SWGS waveguide.

We calculate the effective refractive index (n_{eff}) of silicon-based SWGS waveguide with periodic structures along the propagation using the 3D FDTD method. To guarantee the calculation accuracy, we investigate the effective refractive index of the SWGS waveguide as a function of the mesh resolution (mesh size) at 2100 nm. The results shown in Figure 5A depict that the n_{eff} gets stable when the mesh resolution is less than 30 nm. In the following simulations, we set the mesh resolution as 20 nm to achieve highly accurate results. As shown in Figure 5B, the effective refractive index slightly decreases (1.604–1.553) with the increase of the wavelength (1900–2100 nm).

Figure 5: Effective refractive index as functions of the mesh resolution and wavelength.

(A) Calculated effective refractive index of SWGS waveguide at 2100 nm vs. mesh resolution. (B) Calculated effective refractive index of SWGS waveguide vs. wavelength. The mesh resolution is set as 20 nm.

The proposed SWGS waveguide effectively delocalizes the SWGS mode from the silicon region, holding the potential to greatly reduce nonlinearity. We study the nonlinearity of the SWGS waveguide. In the calculations, the nonlinear refractive indices n_{2} used for silicon and SiO_{2} are 4.5×10^{−18} and 2.6×10^{−20} m^{2}/W [14], respectively. A full-vector model that can weigh the contributions of different materials to the nonlinear coefficient is considered to achieve accurate results.

Using given materials and geometric parameters, one can calculate the effective nonlinear coefficient of the waveguide. The effective mode area is written as follows [15]:

$${A}_{\text{eff}}=\frac{{\left|{\displaystyle \int \mathrm{(}{e}_{v}\times {h}_{v}^{*}\mathrm{)}\cdot \widehat{z}dA}\right|}^{2}}{{\displaystyle \int {\left|\mathrm{(}{e}_{v}\times {h}_{v}^{*}\mathrm{)}\cdot \widehat{z}\right|}^{2}dA}},$$(1)

where *e*_{v} and *h*_{v} are field distributions. The nonlinear coefficient *γ* can be expressed as follows [16]:

$$\gamma =\frac{2\pi {\overline{n}}_{2}}{\lambda {A}_{\text{eff}}}$$(2)

$${\overline{n}}_{2}=k\mathrm{(}\frac{{\epsilon}_{0}}{{\mu}_{0}}\mathrm{)}\frac{{\displaystyle \int {n}^{2}\mathrm{(}x,y\mathrm{)}\text{\hspace{0.17em}}{n}_{2}\mathrm{(}x,y\mathrm{)}\left[2{\left|{e}_{v}\right|}^{4}+{\left|{e}_{v}^{2}\right|}^{2}\right]dA}}{3{\displaystyle \int {\left|\mathrm{(}{e}_{v}\times {h}_{v}^{*}\mathrm{)}\cdot \widehat{z}\right|}^{2}dA}},$$(3)

where *n̅*_{2} is the nonlinear refractive index averaged over an inhomogeneous cross-section weighted with respect to field distribution, *λ* is the wavelength, *k* is the wavenumber, *ε*_{0} is the permittivity of vacuum, and *μ*_{0} is the permeability of vacuum. The effective nonlinear coefficient is finally given as:

$$\overline{\gamma}=\frac{{\displaystyle \underset{L}{\int}\gamma \mathrm{(}z\mathrm{)}dz}}{{\displaystyle \underset{L}{\int}dz}}$$(4)

Figure 6 depicts the calculated effective nonlinear coefficients as a function of the wavelength for strip and SWGS waveguide. It is shown that nonlinearity slightly decreases with the increase of the wavelength. The nonlinearity of the SWGS waveguide at 2100 nm is 4.05/W/m. For comparison, the nonlinearity of the silicon strip waveguide at 2100 nm is 69.1/W/m. As expected, the SWGS waveguide with more light delocalized from the silicon region features much lower nonlinearity.

Figure 6: Calculated effective nonlinear coefficients of strip waveguide and SWGS waveguide vs. wavelength (silicon width: 375 nm, slot width: 100 nm, period: 300 nm, duty cycle: 50%).

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