# Microwave Photonic Frequency Conversion Based on a Wavelength Swept Laser
## 2 Operation Principle
### Simulations of Conversion Performance  
From (7), we know that the conversion coefficient can be tuned by either \(L\) or \(r\). Here, we simulated the trend of the _Sinc_ function in (9), when \(L\) and \(r\) were changed individually, as shown in Figure 2.  
In the simulation, the input frequency was 20 GHz and the conversion coefficient \(\kappa\) was increasing from 0.9 to 1.1 in steps of 0.04. Figure 2a shows the trend of the _Sinc_ function when only \(L\) changed and the absolute value of \(r\) was fixed at 50 GHz/ns. The conversion direction was achieved by switching the sign of \(r\). The tuning of \(L\) was achieved by switching different channels of the optical fiber array. When \(L\) changed, the center frequency of the _Sinc_ function was shifted from 18 GHz to 22 GHz and the 3-dB bandwidth remained the same. Figure 2b shows the trend of the _Sinc_ function when \(L\) was fixed at 10 km and \(r\) was tuned from \(-87\) GHz/ns to 71 GHz/ns. The tuning of \(r\) can be achieved by the dispersion control element or the wavelength sweeping component in the cavity of the WSL. The center frequency shift of the _Sinc_ function was the same as when \(L\) was tuned. However, the 3-dB bandwidth was narrower when \(\kappa\) was closer to 1 and became larger when \(\kappa\) moved away from 1. Thus, the tuning of \(\kappa\) by changing \(L\) was more reliable compared with changing \(r\), especially when the frequency shifting range was large. On the other hand, a smaller chirp rate \(r\) introduced a larger suppression to the sidemodes, as the summation term in (7) describes. Thus, the influence of the pulse repetition frequency \(\omega_{r}\) should also be noted. The pulse repetition frequency \(\omega_{r}\) determined the frequency shift accuracy of conversion system because of the discrete nature of the pulse signal. For the fixed repetition angular frequency \(\omega_{r}\) and conversion coefficient \(\kappa\), increasing \(\tau\) could have significantly narrowed the _Sinc_ function, the sidemodes were suppressed and there was the dominant mode. Figure 3 shows the down-conversion simulation spectra (input: 20 GHz; output: 18.8 GHz) and the up-conversion simulation spectra (input: 20 GHz; output: 21.4 GHz) when \(\omega_{r}\) was set to be 20 MHz and \(\tau\) was increasing. The frequency interval of the longitudinal modes (the blue combs) was 20 MHz. As \(\tau\) increased, the envelope of the _Sinc_ function (the green curves) remained narrow. When the notch period of the _Sinc_ function matched the repetition frequency of the WSL, the side modes could be greatly suppressed. It should be noted that \(\tau\) could not be increased without limitation. When the side modes were mostly suppressed, the duty cycle (\(\tau/T_{r}\)) of the WSL was about 95 %. If \(\tau\) was too large, the adjacent pulses would have interfered with each other. Thus, by adjusting the duty cycle of the WSL, the sidemodes were suppressed. On the other hand, the pulse width \(\tau=\Delta\omega/r\) was related to the chirp rate \(r\) and the bandwidth of the WSL \(\Delta\omega\), and \(r\) will also affect \(\kappa\). We use Figure 4 to deal with the influence of the chirp rate \(r\), the bandwidth \(\Delta\omega\) and the fiber length \(L\) on the frequency conversion efficiency and accuracy. Theoretically, \(r\) and \(L\) had the same effect on \(\kappa\) as (7) described, but their value range was limited in practice. For example, too long an SMF introduced a significant decay of the optical loss. Figure 4a shows the influence of \(r\) and \(L\) on the conversion coefficient \(\kappa\). For a fixed \(L\), \(\kappa\) almost changed linearly with \(r\) at the range of \(-50\) GHz/ns to 50 GHz/ns. For a fixed \(r\), \(\kappa\) decreased linearly with \(L\) when \(r\) was negative, and increased nonlinearly with \(L\) when \(r\) was positive.