## 1 Introduction
In contrast to conventional RISs, the wireless signal incident on the STAR-RIS is divided into transmitted and reflected signals that propagate to both sides of the surface, realizing a fully spatially reconfigurable wireless environment. STAR-RIS consists of a large number of low-cost elements whose transmission and reflection coefficients (TARCs) are adjusted in real-time within the elemental units using control circuits, typically implemented by microcontrollers or field-programmable gate arrays (FPGAs). The output of the optimization algorithm is converted into specific control commands, which are sent to the hardware control circuit to control the STAR-RIS. For passive-lossless design, the elements can possess independent TARCs or coupled TARCs, depending on the electrical and magnetic characteristics of the STAR-RIS. Existing studies on STAR-RIS assume that the phase-shift coefficients for transmission and reflection are independently tunable, presupposing that the electric and magnetic impedances, which are confined to purely imaginary values, can be arbitrarily selected. However, this may not be feasible for entirely passive STAR-RIS [17]. In the recent study referenced as [18], it was demonstrated that within low-cost passive lossless STAR-RIS, there exists an intrinsic coupling between the phase-shift coefficients of transmission and reflection. This interdependency poses novel design challenges, as it necessitates a joint consideration of both transmission and reflection characteristics during the optimization process. The coupling effect complicates the system design but also opens avenues for the development of advanced optimization algorithms and the enhancement of communication system performance. This discovery underscores the need for a holistic approach to the design and analysis of STAR-RIS to fully harness their potential in cost-effective communication networks. To solve the coupling phase shift problem, ref. [19] proposes an efficient cell alternating optimization (AO) algorithm to find high-quality suboptimal solutions. In the work presented by the authors in reference [20], they introduce the AO algorithm. This algorithm performs an exhaustive search over discrete amplitude and phase shift sets, resulting in a high computational complexity. STAR-RIS has significant potential in ISAC technology, which is the fundamental pillar of future 6G. Ref. [21] focuses on outage-based beamforming design for dual-functional radar-communication systems in 6G networks. The beamforming techniques discussed in [22; 23] are related to the beamform optimization in the STAR-RIS-assisted MISO system. Both papers aim to enhance the security and efficiency of wireless communication systems through advanced beamforming strategies. Motivated by the above, we investigate a downlink MISO communication system enhanced by STAR-RIS in this system, in which BS provides service to two legitimate users located around the STAR-RIS, while two illegitimate users are also present. Based on this model, we jointly design an optimization problem for active beamforming (BF) at the BS and passive BF at the STAR-RIS. Our objective is to maximize WSSR while adhering to direct link conditions and quality of service (QoS) constraints. We consider two phasing schemes: the coupled phase (CP) scheme and the independent phase (IP) scheme. In this paper, we focus on maximizing WSSR for the CP scheme. We first approximate the optimal value of WSSR using the Successive Convex Approximation (SCA) method. To address the non-convexity of the QoS constraints, we transform them into solvable convex constraints using the Second-Order Cone Programming (SOCP) method. For the multivariate optimization of the CP scheme, we employ the Block Coordinate Descent (BCD) algorithm, which divides the variables into three blocks and optimizes the beam variables, coupled phase shift variables, and auxiliary variables sequentially in an alternating manner.