# Post-Flood UAV-Based Free Space Optics Recovery Communications with Spatial Mode Diversity
## 4 Methods  
The proposed UAV-based spatial mode diversity FSO system for post-flood recovery communications is illustrated in Figure 1. Four vertical cavity surface-emitting lasers (VCSELs) generating continuous-wave optical signals on the fundamental modes at wavelengths of 850 nm, 880 nm, 910 nm, and 940 nm were structured into eight distinct linearly polarized Laguerre-Gaussian beams, _LP lm_, where \(l\) = 0, 1, 2, 3, 4 and \(m\) = 1, 3. The LG beams were intensity-modulated at 20 Gbps by pre-processed binary data collected from individual sensors on their respective channels. The _LP_ modes and wavelengths used for the respective channels are shown in Table 1. The ground transmitter was enclosed in a case for protection from rain. _LP lm_ modes were generated from the first diffraction order in the Fourier plane using the binarized electric field displayed on individual liquid crystal gratings, as follows. It is known that the Fourier transform of a linear translation is a complex phase shift [49]. Thus, to produce a translation in the Fourier plane, the transverse modal field is first multiplied by a complex phase shift:  
\[\exp[j(\tau_{x}x+\tau_{y}y)] \tag{1}\]  
where \(\tau_{x}\) and \(\tau_{y}\) are linear tilt constants in the horizontal and vertical directions, respectively. The complex tilted field of the _LP_ mode, \(a+jb\), is binarized such that, in regions represented by  
\[\left(b\pm\frac{1}{2}\right)^{2}+a^{2}\geq\ \ \ \frac{1}{4},\ \ \ a\leq 0 \tag{2}\]  
an expanded laser beam is transmitted through a liquid crystal display, and in other regions, no light passes through. The binarized beam may be expressed as a Fourier series expansion [45; 46; 47; 48; 50]:  
\[g(x_{1},y_{1})=m_{o}+\frac{4}{\pi}\ \sum_{n=1}^{\infty}m_{n}\ \ \cos\{n\left[ \xi(x_{1},y_{1})\ +\tau_{x}x_{1}+\tau_{y}y_{1}\right]\} \tag{3}\]  
where \(m_{o}\) is the constant term and \(n\) is the \(n\)-_th_ diffraction order. In the Fourier plane, the diffraction orders are translated linearly and separated:  
\[G(x_{2},y_{2})=M_{o}(x_{2},y_{2})\ \ +\ \sum_{n=1}^{\infty}\left[M_{n}(x_{2}+n \tau_{x},\ \ y_{2}+n\tau_{y})+M_{n}^{*}(n\tau_{x}-x_{2},n\tau_{y}-y_{2})\right] \tag{4}\]  
where \(x_{2}\) and \(y_{2}\) are spatial coordinates in the Fourier plane, \({}^{*}\) is the complex conjugate, and \(M_{n}\) (\(x_{2},y_{2}\)) is the \(n\)-th diffraction order in the Fourier plane. Each binarized field was then Fourier transformed by a 400 cm focal-length achromatic convex lens, and the linearly polarized Laguerre-Gaussian mode, _LP lm_, was extracted from the first diffraction order in the Fourier plane, M1. The transverse modal field distribution of a linearly polarized _LP lm_ mode is expressed as [51]:  
\[E=R^{l}L_{m-1}^{l}(VR^{2})\exp\left(-VR^{2}/2\right)\cos\phi \tag{5}\]  
where \(R\) is the normalized radius, \(\phi\) is the azimuthal angle of the transmitted transverse modal field, \(V\) is the normalized frequency, \(L_{m-1}^{l}\) is the generalized Laguerre polynomial whereby \(l\) is the azimuthal mode number, and \(m\) is the radial mode number. For channel diversity, each sensor transmitted data on two _LP_ modes so that they experience distinct refractive index fluctuations and modal power coupling. For any data stream, the first mode was used as the primary channel, and a second mode was used as the backup channel during adverse weather conditions, such as rain. Different combinations of the azimuthal mode number and radial mode numbers were used for various primary and backup channels, as shown in Table 1. The backup channels operated on different _LP_