## 2 The EFOGSM Scheme  
In this section, the EFOGSM scheme is proposed and introduced.  
Figure 1 shows the system chart of EFOGSM including the modulator and demodulator. The incoming data are split into two groups after serial-parallel conversion. The first group contains data bits \(\log_{2}P_{M}+\log_{2}S_{M/2}\), where \(P_{M}\) represents the primary signal constellations and \(S_{M/2}\) represents the secondary signal constellations. The second group represents spatial bit \(\left\lfloor\log_{2}\sum\limits_{N_{\mathrm{u}}=2}^{Nt}\begin{pmatrix}N_{ \mathrm{t}}\\ N_{\mathrm{u}}\end{pmatrix}\right\rfloor\), where \(N_{\mathrm{u}}\) denotes the number of lasers that activate transmission of the primary and secondary signal constellations, and \(N_{\mathrm{t}}\) denotes the number of lasers used to select the laser subset for transmitting constellation symbol information. The laser subset can vary from activating only two lasers to multiple or all lasers. In all laser subsets, one or more lasers transmit symbols based on the main signal constellation \(P_{M}\), while other lasers transmit symbols based on the secondary signal constellation \(S_{M/2}\). The space bit is used to select the active laser combination. Typically, the transmitted information bits are based on three groups: the primary signal constellation, the secondary signal constellation and the activated laser combination mapping. First, the information  
bits are transmitted through the main signal constellation at a data transmission rate of \(\log_{2}P_{M}\). Second, the secondary signal constellation is used to transmit information bits at a transmission rate of \(\log_{2}S_{M/2}\). Finally, the transmission rate of \(\left|\log_{2}\sum\limits_{N_{u}=2}^{Nt}\binom{N_{t}}{N_{u}}\right|\) can be obtained using the spatial combination of laser and digital modulation constellation. Therefore, the data transmission rate of the proposed EFOGSM can be expressed as follows:  
\[R_{p}^{EFOGSM}=\log_{2}P_{M}+\left|\log_{2}\sum\limits_{N_{u}=2}^{Nt}\binom{N_ {t}}{N_{u}}\right|+\log_{2}S_{M/2} \tag{1}\]  
In the above equation, \(\left\lfloor\ \right\rfloor\) represents rounded down.  
Table 1 illustrates the transport mapping process of EFOGSM. The system uses a 4-pulse amplitude modulation (PAM) modulator for the primary signal constellation and a 2-PAM for the secondary signal constellation. With a transmission rate of \(\mathrm{Rp}=8\) bpcu and \(N_{t}=N_{\mathrm{r}}=4\) (\(N_{t}\) is the number of lasers, and \(N_{\mathrm{r}}\) is the number of photodetectors). The data bits to be transmitted are [101 1110]. The first three bits [101] are transmitted through both the primary and secondary signal constellations. [10] is sent via the primary signal constellation and is equivalent to \(\log_{2}P_{M}\), while the remaining [1] is transmitted through the secondary signal constellation and is equivalent to \(\log_{2}S_{M/2}\). The remaining four bits [1110] are transmitted through the laser index \(\mathrm{L_{1}L_{2}L_{3}}\), the secondary signal constellation bits are transmitted through laser \(\mathrm{L_{1}}\), and the primary signal constellation bits are transmitted through lasers \(\mathrm{L_{2}}\) and \(\mathrm{L_{3}}\). Therefore, the emission vector of EFOGSM is \(\mathbf{x}=\left[S_{M/2}\,\mathrm{P_{M}}\,\mathrm{P_{M}}\,0\right]^{\mathrm{T}}\), which is \(N_{t}\times 1\) in size.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Data Bits**} & **Laser** & **Signal** & \multirow{2}{*}{**DATA BITS**} & **Laser** & **Signal** \\  & **Combination** & **Constellation** & & **Combination** & **Constellation** \\ \hline
0000 & \(\mathrm{L_{1}L_{2}}\) & \(\mathrm{P_{M}\text{-}S_{M/2}}\) & 1000 & \(\mathrm{L_{1}L_{2}}\) & \(\mathrm{S_{M/2}\text{: }P_{M}}\) \\
0001 & \(\mathrm{L_{1}L_{3}}\) & \(\mathrm{P_{M}\text{-}S_{M/2}}\