# Microwave Photonic Frequency Conversion Based on a Wavelength Swept Laser
## 2 Operation Principle  
### Frequency Conversion Principle  
Figure 1 shows the working principle of the proposed microwave photonic frequency conversion system. A wavelength swept fiber laser (WSL) was used as the optical source. The output of the WSL was a series of long pulse trains, the wavelength of which changed linearly over time, periodically. The input microwave signal was then modulated to the WSL output by an optical intensity modulator in order to convert the microwave signal into the optical domain. During the intensity modulation process, the WSL pulse was shaped by the microwave signal, and the microwave signal voltage was converted to the optical intensity, as shown in Figure 1. The modulated optical signal was then injected into a dispersion control element (DCE). Due to the chromatic dispersion characteristic of the broadband WSL, different time delays were introduced to different wavelength components, and the pulse length of the modulated optical signal was stretched or compressed. At the output of the DCE, a photodetector (PD) was used to detect the optical intensity vibration, and the microwave signal was recovered from the optical domain. In this way, the microwave signal was also compressed or stretched, and the frequency of the loaded microwave signal drifted to a high or a low frequency. The direction of the pulse length change depended on whether the sign of the WSL chirp rate and the DCE index were the same or the opposite. When the sign of the WSL chirp rate and the DCE index were the same, the pulse was compressed and the input signal was up-converted. Otherwise, the pulse was stretched and the input signal was down-converted. The compression rate of the optical pulse  
depended on the time delay, which could be calculated by the dispersion of the DCE and the optical bandwidth of the WSL. Therefore, the frequency conversion signal was achieved by changing the parameter of the WSL fiber laser and the dispersion of the DCE. For example, when using the single mode fiber (SMF) as the DCE, the dispersion could be tuned by changing the length of the SMF. The chip parameter of the WSL was also adjustable in many structures [24; 25]. The mathematical analysis is given as follows; the normalized time domain signal of the WSL output can be expressed as:  
\[E(t)=rect\Big{(}\frac{t-nT_{r}}{\tau}\Big{)}\exp[i(\omega_{c}-\pi rt)t]. \tag{1}\]  
where \(rect(x)\) is the rectangular function, \(\omega_{c}\) is the optical center angular frequency, \(r\) is the frequency sweeping chirp rate (frequency shift per time), \(T_{r}\) is the repetition period and \(\tau\) is the pulse width of the WSL. The input microwave signal is:  
\[s(t)=A\cos(\Omega_{0}t). \tag{2}\]  
where \(A\) is the amplitude of the signal and \(\Omega_{0}\) is the angular frequency of the input microwave signal. After the intensity modulation, the output of the modulated optical signal \(E_{m}(t)\) is given by:  
\[E_{m}(t)=E(t)\cdot s(t)=rect\Big{(}\frac{t-nT_{r}}{\tau}\Big{)}\exp[i(\omega_ {c}-\pi rt)t]\cdot[1+A\cos(\Omega_{0}t)]. \tag{3}\]  
where \(m\) is the modulation index. Taking an SMF as the DCE as an example. The dispersion element introduces extra phase to the frequency spectrum of the modulated optical signal. The spectrum of the modulated signal \(E_{d}(\omega)\) after the dispersion effect can be expressed as:  
\[E_{d}(\omega)=E_{m}(\omega)\cdot\exp\Bigg{(}\frac{i\beta_{2}L(\omega-\omega_{ c})^{2}}{2}\Bigg{)}. \tag{4}\]  
where \(\omega\) is the angular frequency, \(\beta_{2}\) is the group-velocity dispersion of the SMF (\(-20.39\) ps\({}^{2}\)/km) and \(L\) is the length of the SMF. The time domain signal from the output of the SMF is given by:  
\[E_{d}(t)=E_{m}(t)\star\mathcal{F}^{-1}\bigg{\{}\exp\Big{(}\frac{i \beta_{2}L(\omega-\omega_{c})^{2}}{2}\Big{)}\bigg{\}} \tag{5}\] \[=\frac{\exp(i\omega_{c}t)\cdotrect\Big{(}\frac{(t-nT_{r})/\tau}{ 1+2\pi\beta_{2}r}t\Big{)}}{\sqrt{1+2\pi\beta_{2}r}L}\Bigg{\{}