## 2 Drone Visible Light Communications with Differentiated LED Beams  
### Drone Visible Light Communications with Lambertian LED Beam  
Initially, the general Lambertian LED beam is utilized to configure the drone's visible light communications. Vitally, radiation intensity is the key metric to measure the spatial radiation characteristics of optical beams. When the LED optical source matches a Lambertian beam, the response radiation intensity \(R_{\text{Lam}}(\phi)\) is given by [32]:  
\[R_{\text{Lam}}(\phi)=\frac{m_{\text{Lam}}+1}{2\pi}\cos^{m_{\text{Lam}}}(\phi) \tag{1}\]  
where \(\phi\) is the irradiance angle from the LED emitter to the light receiver, \(m_{\text{Lam}}\) denotes the Lambertian order. And \(m_{\text{Lam}}\) is given by:  
\[m_{\text{Lam}}=-\frac{\text{In2}}{\text{In}(\cos\phi_{1/2})} \tag{2}\]  
where \(\phi_{1/2}\) is the LED semi-angle. And in Figure 1a,b, the application scenario of drone VLC with this baseline LED Beam for terrestrial and aerial users is shown separately. In a typical drone VLC application scenario of the terrestrial user, the channel gain of the above Lambertian drone VLC \(H_{\text{AZT}}^{\text{Lam}}\) is given as [32]:  
\[H_{\text{AZT}}^{\text{Lam}}=\left\{ \begin{matrix}\frac{A_{\text{PD}}}{A_{ \text{AZT}}^{2}}R_{\text{Lam}}(\phi_{\text{AZT}})T_{s}(\theta_{\text{AZT}})g (\theta_{\text{AZT}})\cos(\theta_{\text{AZT}}),&0\leq\theta_{\text{AZT}}\leq \theta_{\text{FOV}}\\ 0,&\theta_{\text{AZT}}>\theta_{\text{FOV}}\end{matrix} \right. \tag{3}\]  
where \(A_{\text{PD}}\) denotes the detection area of terrestrial user receiver; \(d_{\text{AZT}}\) denotes the distance between the drone LED and the terrestrial user receiver; \(\phi_{\text{AZT}}\) denotes the irradiance angle from the optical source to the terrestrial user receiver; \(\theta_{\text{AZT}}\) denotes the incident angle to the terrestrial user receiver and \(\theta_{\text{FOV}}\) is the field of view (FOV) of receiver, as shown in Figure 1a. And \(T_{s}(\theta_{\text{AZT}})\) is the gain of the optical filter and \(g(\theta_{\text{AZT}})\) is the gain of the optical concentrator, given by [32]:  
\[g(\theta_{\text{AZT}})=\left\{ \begin{array}{ll}\frac{n^{2}}{\sin^{2}(\theta _{\text{FOV}})},&0\leq\theta_{\text{AZT}}\leq\theta_{\text{FOV}}\\ 0,&\theta_{\text{AZT}}>\theta_{\text{FOV}}\end{array} \right. \tag{4}\]  
where \(n\) being the refractive index of the optical concentrator. When a Lambertian optical beam of a drone with altitude \(H\) is applied to configure the aerial to terrestrial (A2T) user link, the A2T geometry relationship of \(\phi_{\text{A2T}}\) and \(\theta_{\text{A2T}}\) could be expressed as:  
\[\cos(\phi_{\text{A2T}})=\cos(\theta_{\text{A2T}})=\frac{H}{d_{\text{A2T}}}= \frac{H}{\sqrt{r_{\text{A2T}}^{2}+H^{2}}} \tag{5}\]  
where \(r_{\text{A2T}}\) being the radial distance between the drone and the terrestrial user on the ground plane. Therefore, for the A2T link, the Lambertian drone channel gain \(H_{\text{A2T}}^{\text{Lam}}\) should be renewed as:\[H_{\text{A2T}}^{\text{Lam}} =\left\{ \begin{array}{cc}\frac{A_{\text{T}}}{d_{\text{A2T}}^{2}} \left(\frac{m_{\text{lam}}+1}{2\pi}\cos^{m_{\text{Lam}}}(\phi_{\text{A2T}}) \right)T_{\text{s}}(\theta_{\text{A2T}})\frac{n^{2}}{\sin^{2}(\theta_{\text{ FOV}})}\cos(\theta_{\text{A2T}}),&0\leq\theta_{\text{A2T}}\leq\theta_{\text{ FOV}}\\ 0,&\theta_{\text{A2T}}>\theta_{\text{FOV}}\end{array} \right., \tag{6}\] \[=\left\{ \begin{array}{cc}\frac{A_{\text{T}}}{d_{\text{A2T}}^{2}} \left(\frac{m_{\text{lam}}+1}{2\pi}\left(\frac{H}{\sqrt{r_{\text{A2T}}^{2}+H^{2 }}}\right)^{m_{\text{lam}}+1}\right)T_{\text{s}}(\theta_{\text{A2T}})\frac{n^ {2}}{\sin^{2}(\theta_{\text{FOV}})},&0\leq\theta_{\text{A2T}}\leq\theta_{\text {FOV}}\\ 0,&\theta_{\text{A2T}}>\theta_{\text{FOV}}\end{array} \right.,\]  
Figure 1: Schematic of 6G drone VLC based on baseline Lambertian light beam for 6G aerial vehicular networks: (**a**) the respective application scenario of terrestrial user, (**b**) the respective application scenario of aerial user.