# Silicon Integrated Nanophotonic Devices for On-Chip Multi-Mode Interconnets
## 2 Silicon Multi-Dimensional (De)Multiplexers
### Silicon Mode (De)Multiplexers
From the curves, one can find a critical point where the effective indices are matched between TE\({}_{0}\) and TE\({}_{1}\) (w\({}_{\text{c,0}}\) = 325 nm, w\({}_{\text{c,1}}\) = 690 nm). The TE\({}_{\text{S}}\) effective index approaches the TE\({}_{0}\) effective index when w\({}_{\text{wg,0}}\) > w\({}_{\text{c,0}}\) and w\({}_{\text{wg,1}}\) < w\({}_{\text{c,1}}\), indicating that the light power is mainly supported by the TE\({}_{0}\) mode in waveguide #0, as shown in the inset of Figure 4b. Around the critical point, both TE\({}_{0}\) and TE\({}_{1}\) modes will be excited in waveguide #0/#1 (see the inset of Figure 4b), indicating the phase matching. The TE\({}_{\text{S}}\) effective index approaches the TE\({}_{1}\) effective index when w\({}_{\text{wg,0}}\) < w\({}_{\text{c,0}}\) and w\({}_{\text{wg,1}}\) > w\({}_{\text{c,1}}\), indicating that the light power is mainly supported by TE\({}_{1}\) mode in waveguide #1, as shown in the inset of Figure 4b. Thus, the input TE\({}_{0}\) mode in waveguide #0 can be gradually \"pushed\" into waveguide #1  
Figure 4: **(a)** The calculated effective indices for TE\({}_{0}\) and TE\({}_{1}\) modes in a SOI waveguide with varied waveguide width (w\({}_{\text{wg}}\)). The inset shows the cross-section of an ADC with some key parameters labeled. **(b)** The calculated effective indices for TE\({}_{0}\) and TE\({}_{1}\) modes (solid lines) and TE\({}_{\text{S}}\)/TE\({}_{\text{AS}}\) super-modes (dashed lines) in an ADC with varied waveguide width (w\({}_{\text{wg,0}}\) and w\({}_{\text{wg,1}}\)). The inset shows the mode profiles. **(c)** The calculated TE\({}_{0}\)–TE\({}_{1}\) coupling ratio for ADC (left panel) and adiabatic ADC (right panel) with varied coupling length. **(d)** The calculated light propagation profiles for ADC (left panel) and adiabatic ADC (right panel) when TE\({}_{0}\) mode is launched. by slowly varying the core widths. The TE\({}_{0}\)-TE\({}_{1}\) mode conversion is also attained in the meantime. To demonstrate this, we calculate the TE\({}_{0}\)-TE\({}_{1}\) coupling ratio with various coupling lengths, as shown in the right panel of Figure 4c. For the conventional ADC, the coupling ratio periodically changes with the coupling length, while for the adiabatic ADC, the coupling ratio monotonically reaches 100% when the coupling length is long enough (\(>\)100 \(\upmu\)m), indicating that the mode conversion is an adiabatic process. One might find some ripples in the coupling-ratio curve, which is mainly induced by the slight interference effect. The light propagation profile is then calculated for the optimized adiabatic ADC, as shown in the right panel of Figure 4d. The complete mode conversion can be observed from the calculated profile. In [38], the silicon adiabatic ADC was demonstrated for the first time. The measurement results show low losses (IL \(<\) 0.3 dB) and low crosstalk (XT \(<\) \(-\)16 dB) over a 100-nm wavelength band. The counter-tapered ADC was proposed and demonstrated in [39] to further enhance the working bandwidth (BW \(>\)180 nm) and relax the fabrication stringency. Later, the taper-etched ADC was proposed as a highly efficient mode (de)multiplexer [40]. Low losses (IL \(<\) 1.3 dB) and low crosstalk (XT \(<\) \(-\)26 dB) were also experimentally demonstrated. However, adiabatic coupling is always accompanied with a long coupling length (see Figure 4c) and a large footprint. Note that the gap width is fixed for the aforementioned ADCs, so the coupling strength is a constant for these structures. The recent studies have shown that a broad bandwidth can be ensured even with quite a short coupling length by simultaneously changing core width (w\({}_{\text{wg,0}}\), w\({}_{\text{wg,1}}\)) and gap width (w\({}_{\text{gap}}\)) [41]. In [42], a short and robust silicon ADC was realized by utilizing the novel STA method. A detailed description of an STA-based coupler can be found in [74].