# Post-Flood UAV-Based Free Space Optics Recovery Communications with Spatial Mode Diversity
## 4 Methods
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Channel/Mode, \(i\)**} & **Mode,** & \multirow{2}{*}{**Signal**} & \multirow{2}{*}{**Type**} & \multirow{2}{*}{**Wavelength**} \\  & _LP lm_ & & & \\ \hline
1 & _LP 01_ & Sensor 1 & Primary & 850 nm \\
2 & _LP 03_ & Sensor 1 & Backup & 850 nm \\
3 & _LP 11_ & Sensor 2 & Primary & 880 nm \\
4 & _LP 13_ & Sensor 2 & Backup & 880 nm \\
5 & _LP 21_ & Sensor 3 & Primary & 910 nm \\
6 & _LP 23_ & Sensor 3 & Backup & 910 nm \\
7 & _LP 31_ & Sensor 4 & Primary & 940 nm \\
8 & _LP 33_ & Sensor 4 & Backup & 940 nm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Channel Characteristics. modes from the primary channels, thus experiencing distinct refractive index fluctuations and mode coupling. The optical signals from all channels were then amplified, multiplexed, aligned, and transmitted to the UAV, which was flown at a height of 7 m above ground, in a straight line toward the ground receiver, located 40 m away from the transmitter. The horizontal distance travelled and height of the flying drone is shown in Figure 1. Following this, the optical beams are reflected from the UAV to the ground receiver. System parameters are provided in Table 2. The beam waist is the location along the propagation direction where the beam radius is minimum. The Rayleigh length is the distance from the beam waist, in the direction of beam propagation, where the beam radius is increased by a factor of the square root of 2. The beam divergence is an angular measure of the increase in beam diameter or radius with distance from the grating. The photodetector responsivity is a measure of optical-to-electrical conversion efficiency of a photodetector. The transverse electric field of the set of eight _LP_ modes used for the transmission may be modeled as:  
\[X=[X_{1}\ X_{2}\ \ X_{3}\ldots\ X_{8}]^{T} \tag{6}\]  
where \(X_{i}\) is the transverse electric field of the transmitted \(i\)-th mode and \(T\) is the transpose of the matrix. The estimated received electric field after propagating through the FSO channel is  
\[Y=[Y_{1}\ \ Y_{2}\ \ Y_{3}\ldots\ \ Y_{8}]^{T}, \tag{7}\]  
where \(Y_{i}\) is estimated electric field of the \(i\)-th received mode and \(T\) is the transpose of the matrix. \(Y\) is related to the channel matrix, \(H\) by [52]  
\[Y=HX+N, \tag{8}\]  
where the channel matrix \(H\) contains the power coupling coefficients,  
\[h_{i,j},\quad i,j\in[1,N] \tag{9}\]  
A charge-coupled device was used to collect the intensity distribution of the received optical signals. The power coupling coefficients were computed for all channels using a modal decomposition method [53]. Multiplicative slow fading and inter-symbol interference from adjacent modes \(i\
eq j\) were assumed. \(N\) is additive noise from the surroundings. The received electric field of the \(i\)-th mode may be estimated as the summation of the individual components of the transmitted electric field, \(X_{j}\)[54]:  
\[Y_{i}=\sum_{j}h_{i,j}X_{j}+N \tag{10}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Beam waist & 0.025 m \\ Operating wavelengths & 850 nm, 880 nm, 910 nm and 940 nm \\ Rayleigh range & 0.5 m \\ Propagation distance & 400 m \\ Transmitted beam diameter & 3 mm \\ Beam divergence & 0.5 mrad \\ Photodetector responsivity & 0.6 A/W \\ Focal lengths of transmitter lens & 40 cm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Systems Parameters. The power coupling coefficient \(h_{ij}\) is the overlap integral between the received transverse electric field, \(Y_{i}\), and transmitted transverse electric field, \(X_{j}\)[55]:  
\[h_{i,j}=\frac{\left|\int\limits_{A}Y_{i}(x,y)\cdot X_{j}{}^{*}(x,y)\ dA\right|^{ 2}}{\int\limits_{A}\left|Y_{i}(x,y)\right|^{2}dA\ \int\limits_{A}\left|X_{j}(x,y)\right|^{2}dA} \tag{11}\]  
where \(A\) is the plane of the transverse electric field. When the set of modes propagate through the atmosphere, it encounters spatially and temporally varying refractive indices, due to random pressure temperature variations.