Vertical slots are used to reduce the nonlinearity of silicon-on-insulator waveguides. In properly designed slot waveguides, approximately 50% of the optical power can be confined in the air-slot and air-cladding region. Compared with a strip waveguide, a factor greater than 15 times more reduction in the real part of nonlinear coefficient can be achieved in the 100-nm wavelength range. In addition, vertical-slot waveguides exhibit large negative chromatic dispersion, which will induce phase mismatch and further suppress nonlinear parametric effects.
©2010 Optical Society of America
Optical strip waveguides are a key basic building block for integrated optical circuits. Such waveguides are used to connect different elements and to form integral parts of larger components, such as on-chip interferometers  and arrayed-waveguide gratings . For many applications, an important characteristic of the waveguide is its ability to maintain a linear transfer function, preventing the optical wave, which contains an analog or digital signal, from being distorted .
There has been increased interest in using silicon as the base material for photonic integrated circuits. However, it is fairly difficult to maintain low nonlinearity in silicon waveguides . A laudable goal would be to modify the strip waveguide in such a way as to reduce the nonlinear coefficient significantly.
The slot waveguide, which exhibits unique field confinement properties, was proposed [5,6] in earlier works. In such a waveguide, the electric field can be highly confined within the low index slot region . Multi-slot waveguides also have been proposed and they have been demonstrated experimentally in recent research [8–10]. As a result of the introduction of multi-slot waveguides, even more electric field can be concentrated within the low index slot region. By choosing a highly nonlinear slot material, it is possible to acquire a waveguide with ultrahigh nonlinearity [11–13].
In our study, single and double vertical slots were used to efficiently reduce the nonlinearity of the on-chip silicon-on-insulator waveguide. By introducing an air slot in the regular strip waveguide, approximately 50% of the optical power can be confined within the linear material (air). The real part of nonlinear coefficient (γre) of the waveguide can be decreased significantly from 120.05 to 16.11 /W/m. Using a double-air-slot of the same total slot width as the single slot, γre can be further reduced to 6.77 /W/m. Compared with strip waveguides of the same sizes, double-vertical-slot waveguides were shown to reduce γre over a 100-nm wavelength range by a factor greater than 15. The vertical-slot waveguides also exhibit large negative chromatic dispersions. This can further suppress nonlinear parametric effects, such as self phase modulation, cross phase modulation, four wave mixing, etc. These waveguides might also be applied in dispersion compensation and tunable optical delays .
2. Waveguide structure and numerical model
Figure 1 shows the cross sections of a strip, a single- and a double-vertical-slot silicon waveguide on a 2-μm silicon dioxide (SiO2) substrate. The low-index slot regions are filled with air, which has a refractive index of one. The slot mode (x-polarized quasi-TE mode) has an electric field parallel to the interface of the silicon waveguide and the SiO2 substrate, and a large fraction of the optical power is confined in the air slot region. As shown in Fig. 1, we consider waveguides with 500-nm height (h) and 500-nm width (w). The term d is the slot width for both the single- and double-vertical-slot waveguides, while s is the spacing between the two slots of the double-vertical-slot waveguide. In our model, the refractive indices of silicon and silica were obtained according to the Sellmeier equation. At 1550 nm, they are 3.4764 and 1.4440, respectively. In the simulation, the nonlinear refractive indices n2 are 4.5 × 10−18 [15,16] and 2.6 × 10−20 m2/W, for silicon and silica, respectively. The two photon absorption (TPA) coefficient (βTPA) of silicon is 7.9 × 10−12 m/W [15,16], while the TPA coefficient of silica is neglected.
Using a full-vector finite element mode solver (COMSOL Multiphysics 3.4), we obtain the electromagnetic fields of the quasi-TE modes in different waveguide structures. A full-vector model  that can weigh the contributions of different materials to the nonlinear coefficient is used to guarantee an accurate result. A nonlinear figure of merit (FOM) defined as the real part of γ divided by 4π times the imaginary part of γ, i.e., γre/(4πγim), is used to characterize the overall nonlinear performance. A widely used nonlinear FOM has been defined as n2/(λβTPA) . Since γre = 2πn2/(λAeff) and γim = βTPA/(2Aeff) in a single, nonlinear material, the nonlinear FOM we use here is equivalent to the widely used nonlinear FOM. Here, Aeff is the effective mode area. By using a perfectly matched layer in the model, the imaginary part of the effective refractive index is obtained to estimate the leakage loss.
3. Field distribution and nonlinearity
The power distributions of the quasi-TE modes in the strip, single-vertical-slot and double-vertical-slot waveguides are shown in Fig. 2 . Almost all of the power of the strip waveguide is located in the silicon region, while a large portion of the power of the slot waveguide is concentrated in the air slot areas. Here, the slot is right in the middle of the waveguide.
The power distributions of the single-vertical-slot waveguide in different materials are shown in Fig. 3(a) . As the slot width increases, the power in Si decreases, while the power in SiO2 increases. The air can hold the maximal amount of power when the slot has a width of 95 nm. Figure 3(b) shows neff and γre as a function of the slot width in a single-vertical-slot waveguide. For air slot waveguides, since part of the light is confined within the low refractive index, linear air region, the effective refractive index (neff) and real part of nonlinear coefficient (γre) of the waveguide decrease as slot width increases. For efficient mode guiding, the effective refractive index of the mode should be greater than the refractive index of the substrate material SiO2 (neff > nSiO2) . Also, as neff approaches nSiO2, the leakage loss through the substrate increases. For a 100-nm single-vertical-slot waveguide, the nonlinearity can be reduced to 16.11 /W/m while maintaining a 1.16 dB/cm leakage loss. Further increasing the slot width gives a larger leakage loss, while the nonlinearity reduction is not significant. A 110-nm single-vertical-slot waveguide has an 8.53 dB/cm leakage loss and γre is 11.11 /W/m. For slot waveguide with low nonlinear coefficient, the effective refractive index is very close to substrate index. Thus, x-polarization only supports the fundamental slot mode.
For a nanometer-scale slot, the fabrication tolerance is an important factor that should be taken into account. Figure 4 shows the effect of the deviation of the slot position on γre for different slot widths in a single-vertical-slot waveguide at 1550 nm. The slashed region indicates neff < nSiO2. From the contour map, we can see that γre increases as the slot position varies. Simulation results indicated that a symmetrical slot structure has the highest amount of light confined within the air. For asymmetric configuration, as the light-field distributions in the two silicon regions are unequal, it gives a reduced light confinement in the air slot region. For larger slot position variation, we can see that γre first increases and then decreases with the slot width. In such a case, one silicon region is prominently larger compared with the other one. Increasing the slot width can first decrease the effective mode area of the larger silicon region and give an enhanced nonlinearity. Also, from the contour map, we can see that a waveguide with a larger slot width has not only lower nonlinearity, but also smaller change of nonlinearity with position variation.
Since the multi-slot waveguide can provide a higher percentage of optical field confinement within the slot region [8–10], we further studied the effect of reducing nonlinearity by using a double-vertical-slot waveguide. Here, we choose a double-vertical-slot waveguide with two 50-nm slots to make a fair comparison with the 100-nm single-slot waveguide. The power distributions of the double-vertical-slot waveguide in different materials are shown in Fig. 5(a) . For the waveguide with a slot spacing of 91 nm, the power in Si reaches its minimal value. Figure 5(b) shows the effect of the slot spacing variation on neff and γre. When the distribution of optical power in silicon reaches its minimum, the effective refractive index and real part of nonlinear coefficient of the double-vertical-slot waveguide also have their minima. A 50-nm double-slot waveguide with a 150-nm slot spacing can achieve a 6.77 /W/m nonlinearity while maintaining a 3.88 dB/cm loss. Triple- or multiple-vertical-slot waveguides can further increase the light confinement in the air region and might further reduce the nonlinearity. Also, larger waveguide size can increase the effective refractive index and thus reduce the leakage loss. A 700 × 500 nm2 waveguide with 3 equally distributed 50-nm slots has a 0.19 dB/cm loss and a 4.66 /W/m γre.
Figure 6 illustrates the real part of nonlinear coefficient as a function of the wavelength. From 1500 to 1600 nm, the nonlinear refractive index of Si increases with wavelength [15,16]. For strip waveguide, as field is highly confined within the Si region, the increase of effective area with wavelength is much less than the change of n2(Si). This induces an increased γre for strip waveguide. However, for slot waveguides, the increment of Aeff is much larger and gives a decreased γre due to the reduced light confinement at longer wavelengths. For a single-vertical-slot waveguide with a 100-nm slot, γre is reduced 80% over a 100-nm wavelength range compared with the strip waveguide. For a double-vertical-slot waveguide with two 50-nm slots and 150-nm slot spacing, the nonlinearity was reduced by a factor of more than 15 (>93%) over the 100-nm wavelength range.
Table 1 summarizes the nonlinearity of the quasi-TE modes in the silicon strip and vertical-slot waveguides at 1550 nm. For the rectangular silicon strip waveguide with a width from 300 to 700 nm, the portion of the optical power that is confined within the silicon is always more than 70%. Consequently, the nonlinearity can be reduced by increasing the effective mode area. One straightforward way to increase the effective mode area is to increase the size of the waveguide. Aeff can be increased from 0.0984 to 0.2728 μm2 by increasing the waveguide width from 300 to 700 nm. Accordingly, the real part of nonlinear coefficient decreases from 288.15 to 66.78 /W/m. However, to reduce nonlinearity by further increasing the size of the waveguide is not a good choice, because the effective mode area does not increase efficiently for larger waveguide. Also, larger waveguides have more higher-order modes. By introducing a 100-nm vertical air slot in the silicon waveguide, one can have only 26.46% of the optical power confined in the silicon. Thus, a further reduction in the γre to 16.11 /W/m is achieved. In addition, by splitting the 100-nm single air slot into two 50-nm air slots, the waveguide has only 19.50% of the optical power in the silicon and gives an even larger amount of light to the air slots. This helps to reduce γre further, to 6.77 /W/m.
Compared with the strip waveguide, vertical-slot waveguides exhibit a large negative chromatic dispersion. From Fig. 7(a) , it is apparent that the absolute value of the chromatic dispersion increases with the width of the slot. For a silicon waveguide with a 100-nm slot, the chromatic dispersion can be as negative as −22.1 ps/nm/m. This large dispersion is helpful in further reducing nonlinear parametric effects, since it can introduce a phase mismatch. Figure 7(b) shows the chromatic dispersion of the double-vertical-slot silicon waveguide. By splitting the 100-nm slot into two 50-nm slots, the value of the absolute dispersion decreases as slot spacing increases. This is due to the reduced waveguide dispersion for larger Si slot spacing in the middle of the whole waveguide. For a double-vertical-slot waveguide with a slot spacing of 100-nm, a dispersion variation of less than 1 ps/nm/m over a 232-nm wavelength range (from 1368 to 1600 nm) can be achieved. Together with a wavelength converter, this negative flat dispersion profile could potentially be used in on-chip tunable optical delay applications .
Vertical slots are used to reduce the nonlinearity of silicon waveguides efficiently. Up to 50% of the optical power can be confined within the low-refractive, air-slot region. Compared with a strip waveguide, a nonlinearity reduction of more than 80% can be achieved over a 100-nm wavelength range using a single-vertical-slot waveguide with a 100-nm air slot. Also, a reduction in γre of more than 93% can be achieved for the 100-nm wavelength range with a double-vertical-slot waveguide. The vertical-slot waveguide also exhibits a large negative chromatic dispersion, which will further suppress the nonlinear parametric effects and find applications in dispersion compensation and tunable optical delays.
This work is sponsored by DARPA Integrated Photonic Delay (iPhoD) program (under contract numbers HR0011-09-C-0124) and HP Laboratories Innovation Research Awards.
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