# Silicon Ribbon-Based Dual-Beam Optical Phased Array with Low Crosstalk and Large FoV
## 2 Theory and Design
From the modulation through the center wavelength 1550 nm to long wavelengths and short wavelengths, the maximum rotation angle by the formula is about \(\pm\)7-8 degrees [13], which will be selected as the design target in this paper. For spot quality improvement after light emission, 20 grating cycles for each period of the chirped SR-based FBG will be utilized to calculate the energy loss through the energy decay theory. The particular decay constant can be expressed in the following Equation (5):  
\[P_{end}=P_{0}e^{-\alpha z} \tag{5}\]  
where \(P_{end}\) is the residual light in the grating, \(P_{0}\) is the energy of input light, \(\alpha\) is the decay coefficient, and \(z\) is the total grating length. Linear radiation in the entire grating length is desired in OPA for good beam spot quality. The inserted SR will enhance the coupling length [12]. From Equation (3), the highest coupling length \(L_{C}\) is around 1575 \(\upmu\)m when the SR width is between 90 nm and 100 nm for a 500 nm transmission waveguide width, as shown in Figure 3a. The SR will cause more coupling, as its width is over 150 nm. Figure 2: (**a**) Dual-beam schematic diagram for a bi-directional FBG dual grating antenna array and the zero-degree top light steering to \(\pm\)0 angles, respectively, for emitting. (**b**) The schematic is demonstrated by adding SR-based FBG. The transmission waveguide with a width lower than 350 nm experiences high optical loss for the non-steady mode and is selected to be between 350 nm and 450 nm for the coupling length optimization after the SR insertion. The following Figure 3b shows the simulation results of the coupling length for various SR widths at the waveguide height of 220 nm under specific transmission waveguide widths. The coupling length is expected to be long to reduce the array crosstalk. The SR grating is inserted in the FBG as a blocking layer between the transmission waveguide array, and its narrow and wide widths are then selected as 90 nm and 100 nm, respectively. We used the FDTD to simulate 12 period FBG on various narrow grating waveguide widths at a fixed 500 nm wide width to calculate the attenuation coefficient, shown in Figure 4. We can conclude that the significant refractive index variation will form stronger radiation with extensive waveguide width changes. Therefore, the light intensity decay in the single grating period length is intense, and the chirped FBG will be applied for the linear decay and steering emission in the FBG. For a conventional single FBG, the most optical power beam radiation typically occurs before half of the effective length of the grating length. The power energy in the FBG behaves like 2 nd or higher-order decay, as shown in Figure 5a. However, the linear dissipation in FBG will form a dense beam spot because the total effective FBG length is proportional to the FWHM of the spectrum. Therefore, the chirped FBG array antenna is chosen and optimized for five periods to obtain the linear optical energy attenuation. Figure 4: The FDTD was used to simulate a 12 period FBG on various narrow grating waveguide widths under a 500 nm wider width for the attenuation coefficient. Figure 3: (**a**) The coupling length, \(\mathrm{L_{C}}\), of the transmission waveguide performs with and without the silicon ribbon addition. (**b**) The SR addition with various widths demonstrates transmission waveguide coupling under specific transmission waveguide widths. These 5 different waveguide widths ranging from 0.36 \(\upmu\)m to 0.45 \(\upmu\)m in a sequence of 0.45 \(\upmu\)m, 0.42 \(\upmu\)m, 0.4 \(\upmu\)m, 0.38 \(\upmu\)m, and 0.36 \(\upmu\)m are selected for the narrowly chirped grating waveguide width and connected to a 500 nm wide transmission waveguide. Due to the lower attenuation coefficient of the higher narrow grating waveguide width, the period number in the 0.5-0.45 \(\upmu\)m grating is higher than in the 0.5-0.36 \(\upmu\)m grating, and the light transmission energy attenuation is shown in Figure 5b.