# Post-Flood UAV-Based Free Space Optics Recovery Communications with Spatial Mode Diversity
## 4 Methods
Spatial modes individually experience different refractive index perturbations, even when propagating in the same path [39]. This causes unique random wavefront aberrations and power spreading for each mode into adjacent modes, evident by the degradation of power coupled into the original mode \(h_{i,j}\). Under atmospheric turbulence, these power fluctuations are slowly time-varying and vary significantly slower than the signal [56; 57]. The Kolmogorov model for turbulent flow is the basis for many contemporary theories and models for emulating atmospheric turbulence for spatial modes [58; 59; 60; 52]. However, these models of turbulence typically only provide statistical averages for the random variations of the atmosphere. Thus, they may be considered insufficient to represent temporal intensity fluctuations and modal crosstalk for a UAV-based system. Hence, in our work, in order to demonstrate the performance of mode diversity for maintaining the signal quality for composite channels, the time-varying optical signal-to-interference-and-noise ratio (SINR) was evaluated under various weather conditions over a time interval of 30 min and sampling interval of 1 s. The optical SINR for the \(i\)-th channel is given by:  
\[ \begin{array}{l}SINR_{i}=\frac{P_{i}}{I_{i}+N}\\ =\frac{\left|h_{i,j}-i\right|^{2}}{\left|h_{i,j}+i\right|^{2}+N}\end{array}  \tag{12}\]  
where \(P_{i}\) is the received power from the \(i\)-th desired mode (\(j=i\)), either in the primary channel or the backup channel. The measured interference power \(I_{i}\) arises from crosstalk from undesirable spatial modes (\(j\
eq i\)). The measured additive noise power, \(N\), constituting thermal noise, ambient light from sunlight, and surrounding objects is minimal compared to the interfering modal power. For optimal signal detection, the ground receiver requires knowledge of the current channel matrix, \(H\). The main challenge lies in the alignment between the ground transmitter to the UAV and the alignment between the UAV to the ground receiver, in order to prevent tip-tilt errors, despite a line-of-sight link between the ground transmitter and ground receiver. To address this issue at the ground transmitter, the conjugates of a known sequence of complex \(LP\) mode fields was transmitted toward the UAV and then reflected back to the ground transmitter to determine the optimum angle for pointing the optical beam from the ground transmitter toward the UAV. Adjustments to the pointing angle were made on the ground transmitter using a rotating shaft, translation stages, and goniometers until the exact LP mode was acquired on the reflected optical signal. Similarly, at the ground receiver, beam tracking of the signals was performed to avoid tip-tilt errors using a rotation shaft, translation stages, and goniometers. Using the conjugate of a known \(LP\) mode, the direction of arrival of the received optical beam was determined for maximizing spatial acquisition. After the optimal angles were determined, the pointing angle and angle of arrivals were fixed, as the UAV travelled in a straight line from the initial point. An alternative method for beam tracking is to measure the displacement offsets of the arriving at the detector, which translate into angles of arrival variations [61]. The SINR threshold \(SINR\gamma\) was set at the minimum level for turbulence-free transmission under heavy rain, such that:  
\[SINR_{\gamma}=\frac{\sum\limits_{i=1}^{8}P_{i}}{\overline{r}_{i}} \tag{13}\]  
where \(\sum_{i=1}^{8}P_{i}\) refers to the sum of the transmitted power in all desired modes and \(\overline{r}_{i}\) is the average receiver noise power per mode. The calculated \(SINR\gamma\) value was 25 dB. As shown in Table 1, two different \(LP\) modes were used, for the primary channel and backup channels, under the same weather condition. The time-varying primary and backup channels were used for obtaining the composite channel, with the aid of a micro-electromechanical systems (MEMS) optical switch for temporal switching between the two channels.