## 2 Principle
### The Impact of CP on the AO-OFDM System  
In the transmitter, the LCoS programmable WSS was used to perform Sinc-shaped filtering on a super-continuum optical source so as to generate multiple orthogonal carriers, which were further modulated to become the sub-carriers of the AO-OFDM system. These sub-carriers were multiplexed by WSS to form the complete AO-OFDM signal. Although the OFDM system with inserted CP required accurate chromatic dispersion compensation and monitoring at the receiving end, the CP could still mitigate the impact of dispersion on the AO-OFDM system, to a certain extent. We used the CP to cooperate with our proposed scheme to improve the system's resistance to dispersion. In order to reduce the influence of lower inter-carrier interference and component defects on the system, the insertion of CP is essential. In a traditional electrical OFDM system, the CP is inserted by adding the end of each symbol to the beginning of the next symbol and approaches functions that can be easily implemented in the AO-OFDM system [10; 17]. In Ref. [10], the CP was added by increasing the filter bandwidth of the filter that generated orthogonal subcarriers by supercontinuum filtering. The signal of the AO-OFDM system with CP can be expressed as:  
\[u_{OFDM}(t)=\sum\limits_{i=-\infty}^{\infty}\sum\limits_{k=0}^{N-1}{{C_{ki}}{u _{k}}(t-iT_{s})} \tag{1}\]  
\[u_{k}(t)=g(t)exp[j2\pi(\frac{k}{T_{s}}+f_{c})t] \tag{2}\]  
Figure 1: AO-OFDM system using Sinc-shaped filters as demultiplexers. On the right are the frequency-domain waveforms of the two filtering schemes and the eye diagrams after transmission under ideal conditions. \[g_{cp}(t)= \begin{cases}1&(-\frac{T_{b}}{2}\leq t<\frac{T_{b}}{2})\\ 0&(other)\end{cases}  \tag{3}\]  
where \(C_{ki}\) is the \(i\)th signal symbol on the kth subcarrier, \(T_{b}\) is the symbol period, \(f_{c}\) is the center frequency of the OFDM system, and \(u_{k}(t)\) and \(g_{cp}(t)\) are the waveform\(-\) and pulse-shaping functions of the \(kth\) subcarrier, respectively. The pulse-shaping function without CP is \(g(t)\):  
\[g(t)= \begin{cases}1&(-\frac{T_{OFDM}}{2}\leq t<\frac{T_{OFDM}}{2})\\ 0&(other)\end{cases}  \tag{4}\]  
where \(T_{s}\) is the symbol period; the frequency domain filter functions corresponding to the two pulse-shaping functions are:  
\[G_{k}(f)=sinc[(f-f_{sk})f_{s}] \tag{5}\]  
\[G_{kcp}(f)=sinc[(f-f_{bk})f_{b}] \tag{6}\]  
The added CP can be calculated as \(cp=(f_{b}-f_{s})/f_{b}\), where \(f_{b}=1/T_{b}\) and \(f_{S}=1/T_{S}\) are the bandwidth of the Sinc function and symbol rate, respectively. \(f_{bk}\) is the center frequency of the Sinc-shaped filter corresponding to the kth orthogonal subcarrier with the filter bandwidth of \(f_{b}\) after inserting CP. \(f_{sk}\) is the corresponding center wavelength without CP being inserted. In the receiver, Sinc-shaped filtering with the same bandwidth as the WSS can be performed on the signal. When the bandwidth \(f_{b}\) of the Sinc-shaped filter is greater than \(f_{s}\), the CP will be inserted into each symbol period, as shown in Figure 2:  
The AO-OFDM signal with CP inserted at the receiving end can be expressed as follows through the corresponding Sinc-shaped filter at the receiving end:  
\[u_{out}(t)=u_{OFDM}(t)*h_{m}(t)=\sum_{i=-\infty}^{\infty}\sum_{k=0}^{N-1}C_{ki} r_{k}(t-iT_{s}) \tag{7}\]  
\[r_{k}(t)=\sum_{m=0}^{N-1}s_{mk}(t)\exp(j2\pi(\frac{k}{T_{s}}+f_{c})t) \tag{8}\]  
\[H_{m}(f)=sinc[(f-f_{bm})f_{s}] \tag{9}\]  
Figure 2: (**A**) The impact of CP on a single orthogonal sub-carrier; (**B**) time-domain pulse envelope of a single symbol without CP being inserted; and (**C**) single time-domain pulse envelope inserted into the CP. \[s_{kk}(t)_{(m=k)}=\left\{ \begin{array}{cc}\frac{1}{T_{s}}\Big{(}t-\frac{-T_{s}-T_ {b}}{2}\Big{)}&(\frac{-T_{s}-T_{b}}{2}\leq t<\frac{-T_{s}+T_{b}}{2})\\ \frac{T_{b}}{T_{s}}&(\frac{-T_{s}+T_{b}}{2}\leq t<\frac{T_{s}-T_{b}}{2})\\ \frac{1}{T_{s}}\Big{(}-t+\frac{T_{s}+T_{b}}{2}\Big{)}&(\frac{T_{s}-T_{b}}{2} \leq t<\frac{T_{s}+T_{b}}{2})\\ 0&(other)\end{array} \right.