# Multi-Beam Focusing and Deflecting Characteristics of Liquid Crystal Optical Phased Array
## 2 Principles and Methods
### GS Algorithm and Simulation  
The GS algorithm is a classical iterative algorithm proposed by [PERSON] and [PERSON] in 1972 [17]. GS algorithms are widely used in optical information processing, especially in tasks such as holographic reconstruction and beam shaping. In liquid crystal optical phased arrays (LCOPAs), the GS algorithm is used to solve the problem of multi-beam formation. It generates the desired phase distribution by iterative calculation from the known target light intensity distribution. This phase distribution is loaded into a liquid crystal optical phased array, resulting in far-field multi-beam deflection.  
We designed an amplitude distribution pattern in which the main lobes were located at some desired angles. Assuming the designed amplitude distribution is labeled as \(D(u)\), and the amplitude distribution of the incident light is \(A(x)\). \(D(u)\) and \(A(x)\) are two constraints in the algorithm.  
The computational steps of the GS algorithm are shown in Figure 2. Firstly, the initial phase \(\varphi_{0}(x)\) and amplitude \(A(x)\) of the incident light are determined. The near-field complex amplitude \(E_{0}(x)\) of the incident light modulated by the LCOPA device is obtained and the Fourier transform is performed to obtain the far-field complex amplitude \(E_{1}(x)\). In the frequency domain, the new complex amplitude distribution \(\hat{E}_{1}(x)\) is obtained by replacing the amplitude with a pre-designed amplitude distribution \(D(u)\), while keeping the phase constant. Then, a new complex amplitude distribution \(\hat{E}_{1}\) is obtained. \(ER\) indicates the root mean squared (RMS) error and \(\varepsilon\) denotes the threshold value. If the cut-off condition is not satisfied, then the inverse Fourier transform is performed to obtain the complex amplitude distribution \(\hat{E}_{0}\) of the incident light. By replacing \(A_{0}(x)\) with \(A(x)\), we proceed to the next iteration. The cycle is iterated until the cut-off condition is satisfied and the best phase value of the incident light \(\varphi_{0}(x)\) is obtained [18].  
According to the GS algorithm introduced above, we simulated the designed phase distribution of muti-beam fan-out and its diffraction characteristics in the far field. We assume that the desired diffraction distribution has two main lobes, and the angles are plus 1.8 mrad and minus 1.8 mrad, which is shown in Figure 3a. The designed phase distribution can be calculated by the above GS algorithm, as is shown in Figure 3b. Based on the [PERSON]'s diffraction formula, the distribution of far-field diffraction optical intensity can be calculated, as is shown in Figure 3c. It can be seen that the two fan-out beams are the two main lobes of the far-field diffraction optical intensity distribution, with angles of plus 1.8 mrad and minus 1.8 mrad, respectively. There are also some side lobes around the two main lobes, which can be reduced by further optimization. Other fan-out profiles can also be simulated by the same method.  
Figure 3: Simulation results of multi-beam fan-out: (**a**) normalized optical intensity distribution desired; (**b**) normalized optical phase distribution calculated by GS algorithm; and (**c**) normalized optical intensity distribution simulated by [PERSON]’s diffraction formula with phase distribution of (**b**).  
Figure 2: Flowchart of the GS algorithm.