# Si Photonics for Practical LiDAR Solutions
## 4 Si Photonics Optical Phased Array
### OPA System Design
#### 4.1.1 Beam Divergence
[FOOTNOTE:†]Footnote †: Note: Ant., antenna design; \(\mathrm{N_{an}}\), number of antennae; \(\mathrm{D_{an}}\), antenna pitch; Beam, beam linewidth; Steer., beam steering range; SLS, sidelobe suppression; PS/St., phase shifter and/or other steering approach; \(\mathrm{P_{\pi}}\), \(\pi\)-shift power; BW, tuning speed or bandwidth. LG, 1D linear grating; LGS, sidewall LG; LGN SiN assisted LG; FG, focusing grating; EE, edge emitting; \(\lambda\), wavelength sweeping; CI, current injection. [FOOTNOTE:†]Footnote †: Note: Ant., antenna design; \(\mathrm{N_{an}}\), number of antennae; \(\mathrm{D_{an}}\), antenna pitch; Beam, beam linewidth; Steer., beam steering range; SLS, sidelobe suppression; PS/St., phase shifter and/or other steering approach; \(\mathrm{P_{\pi}}\), \(\pi\)-shift power; BW, tuning speed or bandwidth. LG, 1D linear grating; LGS, sidewall LG; LGN SiN assisted LG; FG, focusing grating; EE, edge emitting; \(\lambda\), wavelength sweeping; CI, current injection. [FOOTNOTE:†]Footnote †: Note: Ant., antenna design; \(\mathrm{N_{an}}\), number of antennae; \(\mathrm{D_{an}}\), antenna pitch; Beam, beam linewidth; Steer., beam steering range; SLS, sidelobe suppression; PS/St., phase shifter and/or other steering approach; \(\mathrm{P_{\pi}}\), \(\pi\)-shift power; BW, tuning speed or bandwidth. LG, 1D linear grating; LGS, sidewall LG; LGN SiN assisted LG; FG, focusing grating; EE, edge emitting; \(\lambda\), wavelength sweeping; CI, current injection. [FOOTNOTE:†]Footnote †: Note: Ant., antenna design; \(\mathrm{N_{an}}\), number of antennae; \(\mathrm{D_{an}}\), antenna pitch; Beam, beam linewidth; Steer., beam steering range; SLS, sidelobe suppression; PS/St., phase shifter and/or other steering approach; \(\mathrm{P_{\pi}}\), \(\pi\)-shift power; BW, tuning speed or bandwidth. LG, 1D linear grating; LGS, sidewall LG; LGN SiN assisted LG; FG, focusing grating; EE, edge emitting; \(\lambda\), wavelength sweeping; CI, current injection. [FOOTNOTE:†]Footnote †: Note: Note: Ant., antenna design; \(\mathrm{N_{an}}\), number of antennae; \(\mathrm{D_{an}}\), antenna pitch; Beam, beam linewidth; Steer., beam steering range; SLS, sidelobe suppression; PS/St., phase shifter and/or other steering approach; \(\mathrm{P_{\pi}}\), \(\pi\)-shift power; BW, tuning speed or bandwidth. LG, 1D linear grating; LGS, sidewall LG; LGN SiN assisted LG; FG, focusing grating; EE, edge emitting; \(\lambda\), wavelength sweeping; CI, current injection. [FOOTNOTE:†]Footnote †: Note: Note: Ant., antenna design; \(\mathrm{N_{an}}\), number of antennae; \(\mathrm{D_{an}}\), antenna pitch; Beam, beam linewidth; Steer., beam steering range; SLS, sidelobe suppression; PS/St., phase shifter and/or other steering approach; \(\mathrm{P_{\pi}}\), \(\pi\)-shift power; BW, tuning speed or bandwidth. LG, 1D linear grating; LGS, sidewall LG; LGN SiN assisted LG; FG, focusing grating; EE, edge emitting; \(\lambda\), wavelength sweeping; CI, current injection. [FOOTNOTE:†]Footnote †: Note: Note: Ant., antenna design; \(\mathrm{N_{an}}\), number of antennae; \(\mathrm{D_{an}}\), antenna pitch; Beam, beam linewidth; Steer. principle form a well collimated beam without any lenses. Diffraction limit dictates the relationship of the full divergence \(\Theta\) of a beam emitted from an OPA with its effective aperture approximately described by a Gaussian beam model [73]  
\[\Theta\approx 2\frac{\lambda}{\pi w_{0}}\approx\frac{4\lambda}{\pi D}, \tag{4}\]  
where \(\lambda\) is wavelength in transmitting medium, \(w_{0}\) is Gaussian beam waist and \(D\) is the total size of an antenna array in the corresponding FOV direction. Taking the 0.1\({}^{\circ}\) full divergence requirement at 1550 nm for example, the minimal array size, i.e., aperture \(D\), is 1.13 mm and in practical a larger aperture may be needed considering the deviation of the output beam from ideal Gaussian model.