# Silicon Ribbon-Based Dual-Beam Optical Phased Array with Low Crosstalk and Large FoV
## 2 Theory and Design  
For the dual-beam FBG structure, the transmission modes can be formed in the far-field to present two spots, where the effective refractive index \(n_{eff}\) of the grating can be obtained by the effective mode theory and expressed as in the following Equation (1):  
\[n_{eff}=Fn_{I}+(1-F)n_{E} \tag{1}\]  
where \(n_{I}\) and \(n_{E}\) are the effective refractive indices of the silicon-on-insulator (SOI) waveguide in the non-etch and etch parts of the FBG, respectively. \(F\) is the duty cycle of the grating structure and is chosen as 0.5 for easy processing in the linewidth consideration. The distance between the antenna array is an essential parameter to design a large-angle beam scanning angle. The FoV is defined as the angle difference between the main and grating lobes, and the grating lobe angle \(\Phi\) can be expressed in the following formula:  
\[\sin\Phi_{m}=\sin\Phi_{0}+\frac{m\lambda}{d} \tag{2}\]  
where \(m\) is the diffraction order, \(\Phi_{m}\) is the m-order angle of the grating lobe, \(\Phi_{0}\) is the main lobe angle, and \(d\) is the waveguide grating spacing. From the above Equation (2), it can be seen that when \(d\) continues to increase, the angle difference between the main lobe and the first-order grating lobe will decrease, which means that the FoV reduces. Therefore, 600 nm is chosen as the grating spacing to get a large FoV although the crosstalk between the grating waveguides will increase. According to the coupling theory, when two geometrically identical waveguides transmit the same mode of light, it is easy to cause the two transmission modes to affect each other and cause a coupling phenomenon. Therefore, these two transmission modes are coupled entirely when their waveguide length reaches the coupling length [12]. The following Equation (3) can express the coupling length \(L_{c}\) between waveguides:  
\[L_{c}=\frac{\lambda}{2(n_{s}-n_{a})} \tag{3}\]  
where \(n_{s}\) is the symmetrical mode, \(n_{a}\) is the anti-symmetrical mode, and \(\lambda\) is the incident light wavelength. The transmission mode can be regarded as two modes with the same amplitude: even and odd modes. Therefore, the effective refractive index difference will decrease through SR addition between the grating antenna, and the coupling length is enhanced to conduct the mutual crosstalk reduction. Figure 1: Schematic drawing of the SR grating array through a dual-beam optical phased array. In this study, the height of the transmitting silicon waveguide used in SOI is 220 nm, and the width is 500 nm. The center-to-center waveguide pitch of 550 nm is then utilized for a FEM coupling simulation through the effective refractive index difference between symmetric and asymmetric modes of the transmitting waveguides. The coupling length is theoretically derived to optimize the geometric parameters. The effective refractive index of the FBG waveguide with different line width etching degrees was calculated by FEM after adding the silicon strip. The FBG with different line width combinations was designed by the effective refractive index and Bragg diffraction approaches. In this study, dual-beam scanning through bi-directional FBG was achieved by adjusting the wavelengths so that the light emitted from symmetrical angles can be scanned and can coincide at 0 degrees, as shown in Figure 2a. The waveguide grating spacing, d, and the pitch between SR and FBG are demonstrated in Figure 2b. The wavelength angle can be expressed by the following Equation (4):  
\[n_{c}\sin\theta=n_{eff}-\frac{\lambda}{\Lambda} \tag{4}\]  
where \(n_{c}\) is the cladding layer, generally for the air, equal to 1. \(\theta\) and \(n_{eff}\) are the diffraction angle and effective grating index. \(\Lambda\) is the grating period length. The available wavelengths of LiDAR operation are 1500 nm to 1600 nm.