## 2 Hybrid PLC and VLC Channel Based on Distinct Optical Beam Configurations
### Hybrid Channel Based on Distinct Non-Lambertian Optical Beam Configuration  
Different from the well-discussed Lambertian beam, the spatial radiation intensity of non-Lambertian beams provides distinct spatial selectivity for the emitted optical signal [23; 24]. Without a loss of generality, as one representative non-Lambertian optical emission beam with rotational symmetry, the counterpart from the LUXEON Rebel LED was deliberately selected for the investigation of a hybrid PLC and VLC channel in this subsection. The considerations for this choice are that, on one hand, this emission beam presents with quite different spatial radiation characteristics compared with the general Lambertian optical emission beams, while on the other hand, this non-Lambertian emission beam is generated by a commercially available LED, which makes this article applicable for implementation in ubiquitous engineering scenarios. In one representative indoor hybrid PLC and VLC scenario, when the radiation characteristic is introduced from the LUXEON Rebel non-Lambertian beam, the resulting LUXEON non-Lambertian VLC channel gain at the receiver can be written as follows:  
\[H_{\text{VLC}}^{\text{Rebel}}(f)=\eta_{\text{LOS}}^{\text{Rebel}}e^{-j2\pi f \Delta t_{\text{LOS}}}+\eta_{\text{DIFF}}\frac{e^{-j2\pi/\Delta t_{\text{DIFF }}}}{1+jf/f_{C}}, \tag{11}\]  
where \(\eta_{\text{LOS}}^{\text{Rebel}}\) denotes the optical channel gain of the LUXEON non-Lambertian LOS signal. Under this LUXEON non-Lambertian beam pattern configuration, the LOS optical channel gain should be specifically represented by the following:  
\[\eta_{\text{LOS}}^{\text{Rebel}}=\left\{ \begin{array}{cc}\frac{A_{R}}{P_{ \text{normReled}}d^{2}}I_{\text{Rebel}}(\phi)\cos(\theta),&0\leq\theta\leq \theta_{\text{FOV}}\\ 0,&\theta_{0}>\theta_{\text{FOV}}\end{array} \right., \tag{12}\]  
where \(P_{\text{normReled}}\) is the normalization factor for the LUXEON Rebel emission beam, which ensures that the sum power radiated across the whole of the spatial directions is 1 W, and \(I_{\text{Rebel}}(\phi)\) denotes the intensity of the LUXEON Rebel emission beam, which can be profiled by one sum of Gaussian functions [23; 24]:  
\[I_{\text{Rebel}}(\phi)=\sum_{k=1}^{N_{1}}g_{1k}^{\text{Rebel}}\exp\left[-\ln 2 \left(\frac{\left|\phi\right|-\delta_{2k}^{\text{Rebel}}}{\delta_{3k}^{\text{ Rebel}}}\right)^{2}\right], \tag{13}\]  
where \(\phi\) denotes the emission angle, and \(N_{1}=2\) is the number of Gaussian functions. Specifically, the values of coefficients in this expression are as follows: \(g_{11}^{\text{Rebel}}=0.76\), \(g_{21}^{\text{Rebel}}=0^{\circ}\), \(g_{31}^{\text{Rebel}}=29^{\circ}\), \(g_{12}^{\text{Rebel}}=1.10\), \(g_{22}^{\text{Rebel}}=45^{\circ}\), and \(g_{32}^{\text{Rebel}}=21^{\circ}\). Figure 2 illustrates the 3D beam patterns of this LUXEON Rebel non-Lambertian optical beam with rotational symmetry from a side view. Unlike the previous Lambertian optical beam, in this non-Lambertian case, the maximum emission intensity does not appear in the normal direction, i.e., at the direction of the red arrow anymore, but at all directions with an irradiance angle of about \(40^{\circ}\). By substituting (11) into (10), the LUXEON (Lumileds, San Jose, CA, USA) non-Lambertian LOS optical channel gain for this hybrid PLC and VLC can be renewed as:  
\[\eta_{\text{LOS}}^{\text{Rebel}}\ =\ \left\{ \begin{array}{cc}\dfrac{A_{R}}{P_{\text{norm Rebel}}d^{2}}\left\{\sum\limits_{k=1}^{N_{\text{L}}}g_{1k}^{\text{ Rebel}}\exp\left[-\ln 2\left(\dfrac{|\phi|-g_{2k}^{\text{Rebel}}}{g_{3k}^{\text{ Rebel}}}\right)^{2}\right]\right\}\cos(\theta),&0\leq\theta\leq\theta_{\text{FOV}} \\ 0,&\theta_{0}>\theta_{\text{FOV}}\end{array}