## 4 Experimental Results  
We performed two experiments to evaluate the performance of 4-ASK-FOFDM. In the first experiment, the atmospheric turbulence simulator was enabled; thus, it provided a turbulent atmospheric channel with a length of 12.5 m. Before transmitting the 4-ASK-FOFDM signals, we measured the fluctuated signal when a blue light with constant power was transmitted. Because the coherent time of atmospheric turbulence was 1-100 ms, a Si amplified detector (PDA100A-EC, Thorlabs) with bandwidth from DC to 2.4 MHz was used in the measurement. The transmission distance in the water tank was fixed to 1.6 m because the tap water channel did not make the signal fluctuate. The statistical histogram of the captured signal was shown in Figure 4a, and the statistical histogram was fitted according to lognormal distribution; the fit curve was also presented in Figure 4a. Based on the fitted curve, we found that the mean of the captured signal was 0.0313 V, the variance was 0.007, and the scintillation index was 0.007. According to the parameters obtained, we can conclude that the simulated atmospheric turbulence was weak. The power spectra density (PSD) of the fluctuated signal was plotted in Figure 4b. It can be seen from Figure 4b that the power of the fluctuated optical signal was mostly below 19 Hz, which also meant that the simulated turbulence was weak. Comparison of Performance When 4-ASK-FOFDM Signal Transmitted over a Simulated Air Turbulence Channel Followed by a Water Channel or a Free-Space Air Channel  
After the fluctuated channel measurement, the 4-ASK-FOFDM signal was transmitted. To compare with the results obtained in the first experiment, in the second experiment, the atmospheric turbulence simulator was disabled, so it was just a free-space atmosphere channel with a length of 12.5 m. In the two experiments, when the transmission distance through the tap water tank was 3.2 m, we captured the time domain waveforms of 4-ASK-FOFDM. The waveforms and the corresponding spectra are shown in Figure 5. The waveform in Figure 5a and its spectrum in Figure 5b were obtained over the turbulent  
Figure 4: (**a**) Statistical histogram and (**b**) power spectra density of the received signal at the receiver when the light with constant optical power propagated over a simulated turbulence channel. air-water channel in the first experiment. The waveform in Figure 5c and its spectrum in Figure 5d were obtained over the free-space air channel, followed by a water channel in the second experiment. In the two experiments, the effect of the link distance in the water tank on the BER performance was investigated. The link distance through the water tank was changed from 1.6 m to 8 m. The measured BER for 4-ASK-FOFDM at different link distances is shown in Figure 6. As seen from Figure 6, as the link distance increases from 1.6 m to 8 m, the BER of 4-ASK-FOFDM were all below a 7% FEC limit of 3.8 \(\times\) 10\({}^{-3}\), whether the turbulent air channel was enabled or disabled. Furthermore, the difference between 4\({}^{\prime}\)-ASK-FOFDM-turbulent\({}^{\prime}\) and 4\({}^{\prime}\)-ASK-FOFDM-non-turbulent\({}^{\prime}\) was tiny, which meant the turbulent air channel had a negligible effect on the 4-ASK-FOFDM signal. This was because the simulated turbulence by the atmospheric turbulence simulator was too weak, and the power of the fluctuated signal was below 19 Hz; thus, the coherent time of the simulated air turbulence was up to 53 ms. The coherent time was much longer than the period of one OFDM data frame (30 \(\upmu\)s); consequently the turbulent air channel just led to limited fade. In addition, because the IM/DD transmission scheme was adopted, the phase fluctuation brought by the turbulent air cannot affect the 4-ASK-FOFDM signal. Figure 7 shows the constellations of the 4-ASK-FOFDM signal when the link distance through the water tank was 1.6 m and the atmospheric turbulence simulator was either enabled or disabled. The insets in Figure 7 presented the histogram of constellation points.