## 1 Introduction  
Phase arrays have existed for more than a century. Their application in radar in the RF field plays an important role in beam forming and steering. In recent years, autonomous vehicle development has taken advantage of light detection and ranging (LiDAR) for higher spatial resolution. Traditional LiDAR utilizes the mechanical method, such as micro electro-mechanical systems (MEMS) mirrors, to steer the beam, but the inertia from a mechanical device limits the ability to reach a large steering angle and high speed at the same time. On the other hand, the optical phase array (OPA) shows no inertia phenomena, reaches large steering angles at high speed, and has the potential to be integrated into a small foot print.  
The OPA was first developed on the silicon wire-based waveguide array through the optical splitter, phase manipulation, and emitter [1; 2]. OPA through the thermal phase tuning for beam steering showed a lot of applications for solid state LiDAR, sensing, and imaging [3; 4; 5]. Moreover, the unequally spaced waveguide array and different emitter waveguide widths were taken for a large steering angle [6; 7]. For the 180-degree field of view with the grating lob-free beam formation,an index-mismatched waveguide on the emitter with the sub-wavelength pitch was demonstrated by a set of three widths, 300, 350, and 400-nm, in sequence [8]. Since the previous three index-mismatched waveguides own different phases, a conventional approach is to implement addressable control for each emitter phase, which causes the complicated electronic circuit and total power consumption from the independent emitter elements.  
In an OPA architecture, the input light can split into N outputs equally using a cascaded \(1\times 2\) multimode interference (MMI) coupler with n stages, where the relation between N and n is correlated with N = 2\({}^{\text{n}}\). In each stage, only half of the waveguides of each stage are embedded by phase shifters [9]. Then, the accumulated phase shift on the N outputs will be 0, \(\Delta\varphi\), \(2\Delta\varphi\),......, (N-1)\(\Delta\varphi\), respectively. Because the phase shifters on each stage provide the same phase shift, they can be driven by the same voltage. Here, we are using 64 emitters for demonstration. N and n are 64 and 6, respectively. Thus, the total number of voltages for controlling is six. Therefore, we can connect all phase shifters together with one input voltage with each stage, then the total phase shift is 192 \(\Delta\varphi\) (N(log\({}_{2}\)N)/2, N = 64). In order to get an identical phase shift on each stage of N emitters through log\({}_{2}\)N voltages, the phase difference from the index-mismatched waveguides needs to be matched.  
The traditional method of calculating beam steering is phase array theory. The finite difference time domain (FDTD) is used to calculate the intensity distribution of a single antenna and then followed by the phase array theory. FDTD simulation is quite time consuming. In order to improve the computational efficiency on the LiDAR antenna, it is necessary to reduce the simulation time. Therefore, the Rayleigh-Sommerfeld Diffraction (RSD) theory is proposed to calculate the beam formation. The antenna can be treated as a slit from the perspective of Fourier Optics. Then we can quickly obtain the beam steering result through phase manipulation. This beam formation can be from a uniformly/non-uniformly spaced optical phased array through RSD formula under different pitches. Furthermore, the RSD-based genetic algorithm can even be utilized to experimentally facilitate phase distribution.