## 2 Principle
### Analysis of Anti-Dispersion Ability  
We propose the use of a Gauss-shaped filter instead of a Sinc-shaped filter at the transmitting end can improve the system's anti-dispersion capability. As the orthogonal sub-carriers of the WSS--based AO-OFDM system are supercontinuum-filtered by the WSS, the frequency-domain function of the all-optical filter directly controls the frequency-domain waveform of the sub-carrier. The AO-OFDM signal with CP generated by using a Gauss-shaped filter can be expressed as:  
\[u^{\prime}_{OFDM}(t)=\sum_{i=-\infty}^{\infty}\sum_{k=0}^{N-1}C_{ki}u^{\prime}_ {k}(t-iT_{s}) \tag{12}\]  
\[u^{\prime}_{k}(t)=g^{\prime}_{cp}(t)e^{2\pi(\frac{k}{T_{s}}+f_{c})t} \tag{13}\]  
\[g^{\prime}_{cp}(t)=\sqrt{\frac{\pi}{2}}\cdot exp(-\frac{\pi^{2}t^{2}}{2T_{s}^ {2}}) \tag{14}\]  
where \(u^{\prime}_{k}(t)\) and \(g^{\prime}_{cp}(t)\) are the waveform and pulse-shaping function of the \(k\)th subcarrier. The frequency domain waveform of the Gauss-shaped filter is:  
\[G^{\prime}_{kcp}(f)=exp\Bigg{[}-\frac{2(f-f_{kk})^{2}}{f_{s}^{2}}\Bigg{]} \tag{15}\]where \(f_{hk}\) is the center frequency of the Gauss-shaped filter corresponding to the \(k\)th subcarrier with the filter bandwidth of \(f_{s}\) after inserting CP. Different from using a Sinc-shaped filter to insert the CP, the Gauss-shaped filter increased the sub-carrier spacing without increasing the filter bandwidth. As shown in Figure 3, the Gauss-shaped filter was highly similar to the Sinc-shaped filter with a bandwidth of \(f_{s}\) when the intensity was greater than \(-6\) dB. The left axis in Figure 3 shows the amplitude response of the two filters' waveforms with frequency, and the right axis shows the phase response curve of the two filters with frequency. The blue dotted line represents the Gauss-shaped filter and the red solid line represents the Sinc-shaped filter. We first analyzed the orthogonality of the AO-OFDM system based on a Gauss-shaped filter under ideal conditions without being affected by other noises. Obviously, the carrier generated by the Gauss-shaped filter did not have strict orthogonality. There was crosstalk between carriers and crosstalk between symbols at the receiving end through the Sinc-shaped filter. The AO-OFDM signal after passing through the Sinc-shaped filter at the receiving end can be expressed as:  
\[u^{\prime}_{out}(t)=u^{\prime}_{OFDM}(t)*h_{m}(t)=\sum_{i=-\infty}^{\infty}\sum _{k=0}^{N-1}C_{k}r^{\prime}_{k}(t-iT_{s}) \tag{16}\]  
\[{r_{k}}^{\prime}(t)=\sum_{m=0}^{N-1}s^{\prime}_{mk}(t)\exp\biggl{[}j2\pi( \frac{k}{T_{s}}+f_{c})t\biggr{]} \tag{17}\]  
\[H_{m}(f)=sinc[(f-f_{bm})f_{s}] \tag{18}\]  
\[s^{\prime}_{kk}(t)_{(m=k)}= \begin{cases}\frac{1}{2}\cdot erf\Bigl{[}\frac{\pi }{\sqrt{2}T_{s}}(t+\frac{T_{s}}{2})\Bigr{]}-erf\Bigl{[}\frac{\pi}{\sqrt{2}T_{s }}(\frac{T_{s}}{2}-t)\Bigr{]}&(t\leq-\frac{T_{s}}{2})\\ \frac{1}{2}\cdot erf\Bigl{[}\frac{\pi}{\sqrt{2}T_{s}}(t+\frac{T_{s}}{2}) \Bigr{]}+erf\Bigl{[}\frac{\pi}{\sqrt{2}T_{s}}(\frac{T_{s}}{2}-t)\Bigr{]}&(- \frac{T_{s}}{2}<t\leq\frac{T_{s}}{2})\\ \frac{1}{2}\cdot erf\Bigl{[}\frac{\pi}{\sqrt{2}T_{s}}(t+\frac{T_{s}}{2}) \Bigr{]}-erf\Bigl{[}\frac{\pi}{\sqrt{2}T_{s}}(\frac{T_{s}}{2}-t)\Bigr{]}&(t> \frac{T_{s}}{2})\end{cases}  \tag{19}\]  
\[s^{\prime}_{mk}(t)_{(m\
eq k)}=\frac{j}{2}\cdot\exp\biggl{\{}\frac{-\Bigl{[}2 (m-k)^{2}f_{s}^{2}\Bigr{]}}{f_{s}z^{2}}\biggr{\}}\biggl{\{}erf\Bigl{[}\frac{4( m-k)^{2}f_{s}^{2}+j\pi f_{s}(2f_{s}t-1)}{2\sqrt{2}f_{s}}\Bigr{]}-erf\Bigl{[} \frac{4(m-k)^{2}f_{s}^{2}+j\pi f_{s}(2f_{s}t+1)}{2\sqrt{2}f_{s}}\Bigr{]} \biggr{\}} \tag{20}\]  
The Sinc-shaped filter used at the receiving end of the two AO-OFDM systems is the same. If the same-sized CP is inserted, the filter function in the frequency domain is \(H_{m}(f)\), and \(r^{\prime}_{k}(t)\) and \(s^{\prime}_{mk}(t)\) are the waveform and pulse-shaping function of the \(k\)th subcarrier,  
Figure 3: Comparison of the Sinc-shaped filter (red lines) and Gauss-shaped filter (blue lines) in the frequency domain. respectively.