## 2 Theory  
The phase array theory is typically adopted for the far-field optical intensity distribution as the beam formation, which requires the FDTD calculation from a single emitter, shown in Equation (1). FDTD simulation requires high performance computing hardware. In order to improve the LiDAR beam steering efficiency, an algorithm from the scalar diffraction theory--RSD formula--is proposed to manipulate the correct phase adjustment for the optical intensity distribution when an emitter can be treated as a slit. Furthermore, there is no need for an FDTD-assisted phase array theory to illustrate the far-field intensity.  
In order to understand the theoretical phase array principle, a one-dimensional array consisting of N emitters was allocated along the \(z\)-axis of the spherical coordinates [\(r\), \(\theta\), \(\phi\)], and the emitter position showed the individual phase and field amplitude as and \(\psi_{n}\) and \(A_{n}\), respectively. Assuming that the emission profiles from the individual emitters are the same, each emitter can be expressed as a single function \(E(\theta,\ \phi)\). The function \(E(\theta,\ \phi)\) is called the element factor, which is the emission field distribution from a single emitter. Assuming that the phase array has uniform spacing and amplitude, the electric field distribution of the far field \(E_{\text{total}}\) can be expressed as Equation (1). [10; 11; 12]  
\[E_{\text{total}}(\theta,\ \phi)=\sum_{\text{n}=0}^{\text{N}-1}\text{e}^{\text{i} \mskip 1.0 mu k_{\text{n}}\cos\theta}\big{(}A_{\text{n}}\text{e}^{\text{i} \mskip 1.0 mu k_{\text{n}}E(\theta,\ \phi)}\big{)} \tag{1}\]  
where \(k=2\pi/\lambda\), \(r_{\text{n}}\) is the position for each emitter, and N is the emitter number.  
Since this element factor should be derived through FDTD, the mode propagation for a longer emitter, such as 100 \(\upmu\)m, will require a significant amount of calculation time. The beam formation can be treated as the diffraction pattern, and the input pattern is decomposed into its constituent spatial frequencies by Fourier transform. Then, the propagated field of each spatial frequency component can be calculated. The desired diffraction pattern of the given input pattern is the sum of the propagated patterns of the spatial frequency components [13].  
However, the RSD theory could treat the optical phase array as a number of emitters injected by the optical source with various dedicated phases. According to the Huygens-Fresnel principle in the slit diffraction of Fourier optics, every wavefront point is a spherical point source for the next wavefront. The light from the emitter array forms an interference pattern in the far field and could be treated as multiple waveguide-shape slits by basic assumption. Each slit has a different distance from the origin and has a specific phase \(\Theta\), which is expressed in the following [14]:  
\[\Theta=kd_{\mathrm{i}}\cos\theta \tag{2}\]  
where the \(d_{\mathrm{i}}\) is the distance of the i-th emitter from the origin of the array, \(\theta\) is the steering angle between 0\({}^{\circ}\) and 180\({}^{\circ}\), and \(k=2\pi/\lambda\).  
The Rayleigh-Sommerfeld Diffraction formula is from the input field Fourier transform and then used to derive the Fresnel-Kirchhoff's integral equation, as shown in the following [13]:  
\[E(x_{\mathrm{i}},\ y_{\mathrm{i}})=\frac{z}{j\lambda}\ \iint E(x_{\mathrm{o}},y_{ \mathrm{o}})\frac{e^{\mathrm{j}\left(\left(x_{\mathrm{i}}-x_{\mathrm{o}} \right)^{2}+\left(\ y_{\mathrm{i}}-y_{\mathrm{o}}\right)^{2}+z^{2}\right)^{ \frac{1}{2}}}}{\left(x_{\mathrm{i}}-x_{\mathrm{o}}\right)^{2}+\left(y_{\mathrm{ i}}-y_{\mathrm{o}}\right)^{2}+z^{2}}dx_{\mathrm{o}}dy_{\mathrm{o}} \tag{3}\]  
where \(x_{\mathrm{o}}\) and \(y_{\mathrm{o}}\) are object planes, and \(x_{\mathrm{i}}\) and \(y_{\mathrm{i}}\) are image planes. \(z\) is the vertical distance between the object planes and the image planes, and \(\mathrm{d}x_{\mathrm{o}}\)\(\mathrm{d}y_{\mathrm{o}}\) is the small area on the object plane.