## 3 MDM-FSO System Simulation Setup Model  
Figure 4 illustrates the proposed secure MDM-based modified duobinary free-space optics transmission system, modeled using OptiSystem 19.0 software. The system employs three MDB modulation channels, each carrying a 40 Gbps, 40 GHz signal, which are multiplexed onto three laser modes: LP\({}_{01}\), LP\({}_{11}\) mode, and LP\({}_{02}\). The combined signal is transmitted through a 13 km three-mode fiber, followed by a 1.1 km free-space optics (FSO) channel. A SCW laser is used to excite the three LP modes, as depicted in Figure 4.  
Figure 3: (**a**) MDB-MDM transmitter model. (**b**) Optical spectrum. The pseudo-random bit sequence (PRBS) generator produces 40 Gbps of data, which is subsequently modulated using an NRZ/RZ generator and a duobinary precoder with a delay of \(-1\) bit. These MDB signals are processed through an electrical subtractor and further modulated at 7.5 GHz using a sine wave generator operating at 40 GHz. This setup demonstrates the capability of MDM-MBD-FSO systems to achieve high data rates and secure optical communication over both fiber and free-space channels, highlighting its potential for advanced optical communication applications. The MDB signal is modulated using two LiNbO\({}_{3}\) modulators, driven by a SCW source on the LP modes. The signal is then transmitted through a three-mode fiber and subsequently over an FSO link. The FSO transmitter and receiver apertures are optimized to 20.2 cm and 16.3 cm, respectively, ensuring efficient signal transmission and reception. Following FSO transmission, a 40 GHz signal is introduced after the spatial photodiode for the final down-conversion of the signal. A decoder is then employed to demodulate the resulting electrical signal, enabling accurate data recovery. Figure 5 illustrates the three LP modes generated at the output, visualized using an optical visualizer, showcasing the distinct mode patterns achieved through the system. Figure 4: Proposed hybrid MDM-FSO link. Figure 5: An LP mode exited by the spatial CW laser (**a**) LP\({}_{01}\) (**b**) LP\({}_{11}\) (**c**) LP\({}_{02}\). The key parameters adopted for the three-mode fiber model used in this study are the core radius (a), cladding radius (b), core refractive index (n1), cladding refractive index (n2), and operating wavelength (\(\lambda\)). These features are crucial for calculating the normalized frequency (V-number), which determines the number of propagating modes in the fiber. Table 1 summarizes some of the parameter values used for modeling the three-mode fiber. The FSO link is similarly modeled using the parameters depicted in Table 2 under specific predefined atmospheric conditions. The performance of the FSO communication link is verified by comparing the results of the proposed design with the measurement standards established by the Kim model [24]. According to the Kim model, the acceptable minimum binary error rate in telecommunications is \(10^{-9}\), and the value of Quality Factor Q is approximately six [25]. The link is modeled based on the assumption that thermal noise and background radiation noise are the primary noise sources. Ignoring signal-related noises, the relevant noise terms are presented in Equation (3). \[\sigma^{2}=\sigma_{th}^{2}+\sigma_{shot}^{2}+\sigma_{bxb}^{2} \tag{3}\]  
The concepts of background light and background light beat noise are denoted by \(\sigma_{bxb}^{2}\), alongside the distinct representation of thermal noise \(\sigma_{th}^{2}\) and shot noise \(\sigma_{shot}^{2}\). Together, these elements play a crucial role in understanding the overall noise conditions. \[\sigma_{th}^{2}=4 KTB_{e}/R_{L} \tag{4}\]  
\[\sigma_{short}^{2}=2 qI_{bxb}B_{e} \tag{5}\]  
\[I_{b\times b}=RN_{b}B_{O} \tag{6}\]  
_Be_ denotes the electronic bandwidth of the system, \(K\) denotes the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(R_{L}\) represents the photodetector load resistance.