# Microwave Photonic Frequency Conversion Based on a Wavelength Swept Laser
## 2 Operation Principle
### Frequency Conversion Principle
\begin{array}{ c}\exp\Big{(}-\frac{i\pi t^{2}}{1+2\pi\beta_{2}r}t\Big{)}\\ +\frac{4}{2}\exp\Big{(}-\frac{i(\Omega_{0}-2\pi r)t^{2}}{8\pi\beta_{2}r}t+4 \pi r\Big{)}+\frac{i\Omega_{0}^{2}}{4\pi r}\Bigg{\}}.\end{array} \]  
Figure 1: The working principle of the proposed microwave photonic frequency conversion system (the red arrows are the optical path; the blue arrows are the electrical path; the curves with the gradient face color are the optical pulses; the blue curves are the microwave signals; RF\({}_{\text{in}}\) is the input radio frequency signal; RF\({}_{\text{out}}\) is the output radio frequency signal). where \(\mathcal{F}^{-1}\) denotes the inverse Fourier transform. At the output of the PD, the high frequency optical components cannot be detected; thus, the photocurrent signal is given by:  
\[ \begin{array}{c}I_{PD}(t)=\Re\cdot E_{t}^{*}(t)\cdot E_{d}(t)\\ =\frac{\Re\cdot\text{{rec}}\left(\frac{(t-nT_{r})/t}{1+2\pi\beta_{2}rL}\right)} {1+2\pi\beta_{2}rL},\\ +2A\cos\left(\frac{2\Omega_{0}t}{1+2\pi\beta_{2}rL}\right)\cos\left(\frac{\beta _{2}L\Omega_{0}{}^{2}}{2+4\pi\beta_{2}rL}\right)\end{array} \right\}. \tag{6}\]  
where \(\Re\) is the sensitivity of the PD. Here, we define:  
\[\kappa=\frac{1}{1+2\pi\beta_{2}rL} \tag{7}\]  
as the conversion coefficient. By adjusting the product of \(r\) and \(L\), the conversion coefficient can be tuned. When \(\beta_{2}\) and \(r\) have opposite signs, the signal is compressed. Otherwise the signal is stretched. Then, (6) yields:  
\[ \begin{array}{c}I_{PD}(t)=\Re\kappa\cdot\text{{rec}}\left(\frac{(t-nT_{r}) \kappa}{\tau}\right)\left\{\begin{array}{c}1+\frac{A^{2}}{2}+\frac{A^{2}}{2} \cos(2\kappa\Omega_{0}t)\\ +2A\cos(\kappa\Omega_{0}t)\cos\left(\frac{\beta_{2}L\kappa\Omega_{0}{}^{2}}{2 }\right)\end{array}\right\}\end{array}  \tag{8}\]  
In small signal modulation conditions, the \(A^{2}\) terms can be ignored and the constant term in brackets is the low frequency component which can be easily filtered. The electrical spectrum of the microwave signal after conversion can be expressed as:  
\[ \begin{array}{rl}S_{c}(\omega)&=[I_{PD}(t)]=\left[\Re\kappa\cdot\text{{rec}} \left(\frac{(t-nT_{r})\kappa}{\tau}\right)\left\{2A\cos(\kappa\Omega_{0}t) \cos\left(\frac{\beta_{2}L\kappa\Omega_{0}{}^{2}}{2}\right)\right\}\right]\\ &=\mathcal{F}\left[\Re\kappa\cdot\text{{rec}}\left(\frac{(t-nT_{r})\kappa}{ \tau}\right)\right]\ast\mathcal{F}\left[2A\cos(\kappa\Omega_{0}t)\cos\left( \frac{\beta_{2}L\kappa\Omega_{0}{}^{2}}{2}\right)\right]\\ &=\left[\frac{\Re\kappa}{2\pi T_{r}}\cdot\text{{sinc}}\left(\frac{\gamma \omega}{\kappa}\right)\sum\limits_{n=0}^{+\infty}\delta(\omega-n\omega_{r}) \right]\ast\left[2A\cos\left(\frac{\beta_{2}L\kappa\Omega_{0}{}^{2}}{2} \right)\delta(\omega-\kappa\Omega_{0})\right]\\ &=\frac{2A\Re\tau}{2\pi T_{r}}\cos\left(\frac{\beta_{2}L\kappa T_{0}{}^{2}}{2 }\right)Sint\left(\frac{\gamma\omega}{\kappa}\right)\sum\limits_{n=0}^{+ \infty}\delta(\omega-n\omega_{r})\ast\delta(\omega-\kappa\Omega_{0})\\ &=\frac{\Re\kappa}{2\pi\Omega_{r}}\cos\left(\frac{\beta_{2}L\kappa\Omega_{0}{} ^{2}}{2}\right)Sint\left(\frac{\gamma(\omega-\kappa\Omega_{0})}{\kappa}\right) \sum\limits_{n=0}^{+\infty}\delta(\omega-n\omega_{r})\end{array}  \tag{9}\]  
where \(\mathcal{F}\) denotes the Fourier transform, \(\ast\) denotes the convolution and \(\omega_{r}=2\pi/T_{r}\) is the repetition angular frequency of the WSL. The first term (with the cosine function) in (9) can be considered as an attenuation coefficient that relates to the input microwave frequency and the conversion coefficient. The last summation term shows that the spectrum is discrete, due to the pulsed characteristic of the signal. The middle term is a _Sinc_ function (the Fourier transform of the rectangular function) with the center frequency being \(\kappa\Omega_{0}\). The amplitude of the _Sinc_ function decayed as it moved away from the center frequency. Thus, most of the spectrum energy was constrained to \(\kappa\Omega_{0}\) and the input signal was shifted to \(\kappa\Omega_{0}\) successfully.