Free-space optical (FSO) communications provide higher bandwidth and immunity to electromagnetic interference compared to radio communications. To increase FSO system capacity, space-division multiplexing (SDM) with orbital angular momentum (OAM) modes has been explored. However, atmospheric turbulence induces power leakage among modes, leading to significant intermodal interference and performance degradation. Efficient power allocation and mode selection are essential to mitigate these effects. This paper addresses the joint mode selection and power allocation (JMSPA) problem for optimizing data transmission in OAM-multiplexed FSO systems in the presence of atmospheric turbulence. The proposed algorithm extends the Generalized Linear Fractional Programming (GLFP) approach by leveraging its polyblock approximation and projection framework while introducing key modifications to handle the mixed-integer nonlinear fractional structure of JMSPA. Unlike standard GLFP, JMSPA employs a specialized projection step based on a bisection-based search, enabling efficient resolution of the nonlinear rate constraints within the max-min weighted rate (WR) problem. Additionally, instead of a single polyblock approximation, JMSPA constructs distinct polyblocks for each mode subset, effectively managing the mode selection constraints and ensuring a globally optimal solution. The performance of the proposed algorithms is evaluated through channel simulations under different turbulence levels. Results demonstrate significant link performance improvements over uniform power allocation, with notable gains in stronger turbulence conditions.

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