## 3 Results and Discussion  
Here, two index-mismatched silicon wires with 400 and 500 nm in width are proposed to illustrate the beam formation from the emitter pitch of 775-nm (\(\lambda\)/2), without any evanescent coupling blocking between emitters. Instead of addressable thermal tuning control, the waveguide width on the 400-nm side will be implemented by an adiabatic taper, which phase contribution could be derived from its various effective index through the integral form, shown in Figure 1a. The \(\varphi_{1}\) is the phase from 0.5-\(\upmu\)m width at 824.7-\(\upmu\)m length and \(\varphi_{2}\) phase from 0.4-\(\upmu\)m width at 824.7-\(\upmu\)m length. The \(\varphi_{3}\) is the phase from 0.5-\(\upmu\)m width at 10-\(\upmu\)m length. On the other hand, \(\varphi_{4}\) is the phase transferred from 0.5-\(\upmu\)m to 0.4-\(\upmu\)m width at a 10-\(\upmu\)m taper length. Finally, the total phase difference including 12.4 \(\upmu\)m part between the two index-mismatched silicon wires is \(2\pi\), which can be calculated from the listed parameters shown in Figure 1a. The FDTD simulation is applied on two index-mismatched array emitters with a 775-nm spacing through a set of five widths (500, 400, 500, 400, 500 nm at -1.55, -0.775, 0, 0.775, and 1.55 \(\upmu\)m, respectively) and the propagation length is 800 \(\upmu\)m, shown in Figure 1b. When the optical power is injected into the center on the left side, there is almost no evanescent coupl  
Figure 1: (**a**) Two index-mismatched silicon wires-based emitter array (**b**) FDTD simulation. propagates only in the central waveguide it is injected into. In Figure 1b, the central waveguide power will not couple to the adjacent 4 waveguides at a 775-nm pitch for two index-mismatched silicon wires of 400-nm and 500-nm in width. When the emitter output modes are different, the grating lobe can be observed and shown in Figure 2a. After the additional adiabatic tapers are added as 12.4 \(\upmu\)m and the output waveguide modes from 400- and 500-nm in width are the same, the grating lob-free can be demonstrated again as shown in Figure 2b, and the total phase difference can be maintained as 2\(\uppi\). For most LiDAR applications, a full width at half maximum (FWHM) is referred as the angular resolution, and it is the most important indicator along with side mode suppression ratio (SMSR). The optical phase array emitters are spaced much farther apart than the desired half-wavelength pitch and the performance tradeoff is between the field of view and beamforming efficiency. The sub-wavelength spacing between emitters causes strong evanescent coupling and induces the grating and side lobes. On the other hand, the large emitter pitch will generate the grating lobes to share the main beam power, significantly drop the steering efficiency, and lower the field of view. Typically, the FWHM is smaller with an increased waveguide number and higher SMSR can also be applied [11]. When the equal spacing emitter with a 64-emitter array is considered, the RSD is taken to simulate the SMSR and FWHM versus the full steering angles at the different emitter pitch, d, as shown in Figure 3a,b, respectively. In order to obtain a larger SMSR and better FWHM angular resolution, six different emitter pitches, 0.5-\(\lambda\), 0.6-\(\lambda\), 0.7-\(\lambda\), 0.8-\(\lambda\), 0.9-\(\lambda\), and 1.0-\(\lambda\), are taken for simulation. When the pitch is increased, the SMSR will get worse at fixed steering angles, but the FWHM is becoming smaller for higher angular resolution. The SMSR under the pitch of 1.0 \(\lambda\) drops to the minimum value at 60 degrees of the full degree angle, and will be enhanced to 13-dB when the pitch is reduced to 0.5 \(\lambda\). A FWHM at 0.5 \(\lambda\) increases 0.5 degrees compared with 1.0 \(\lambda\) at 140 degrees of the full degree angle. The small pitch without the crosstalk between the emitter array is the highly demanding criterion for the beam steering.