## 2 Basic Theory
### The Double Sideband with Carrier Suppression (DSB-CS) Modulation Method
\begin{array}{l}P_{MZMout}(t)\propto lim_{T\rightarrow\infty}\int\limits_{-T}^{T} \frac{1}{2T}\big{|}E_{MZMout}(t)E_{MZMout}^{*}(t)\big{|}dt\\ \propto\left[\begin{array}{c}E_{0}^{2}j_{1}^{2}(m)+E_{0}^{2}j_{-1}^{2}(m)+2E_ {0}^{2}J_{1}(m)J_{-1}(m)\cos(j2\omega_{RF}t+j2\varphi)+\\ 2E_{0}^{2}J_{1}(m)J_{-1}(m)\cos(j2\omega_{c}t+j\pi)\end{array}\right]\end{array}  \tag{5}\]  
where the first two items, \(E_{0}^{2}j_{1}^{2}(m)\) and \(E_{0}^{2}j_{-1}^{2}(m)\), are the spectral response terms that represent the DC components. The last two terms are the time-varying response terms of optical power, which are related to the frequency response of the photodetector. After collection and beating at the receiver side, the microwave carrier with the angular frequency of (\(2\omega_{RF}\)) is generated. Obviously, by changing the frequency (\(\omega_{RF}\)) of the RF signal, which can be achieved when driving the MZM at the transmitting side, a tunable microwave carrier with doubling-frequency (\(2\omega_{RF}\)) can be successfully achieved at the receiver side. Specifically, through the adjustment of the driving voltage (\(V_{RF}\)) corresponding to \(J_{1}(m)J_{-1}(m)\) or the input light power corresponding to \(E_{0}^{2}\), the same \(P_{MZMout}(t)\) can be realized regardless of the differences between MZMs working in different wavelength regions, indicating the wavelength-transparent property. Therefore, if the wavelength dependance of the detector is not considered, the microwave carrier generated by the DSB-CS modulation method would demonstrate prop erties involving independence from the optical carrier wavelength, frequency doubling, and tunability. RF Power Generation Efficiency Based on the Indium-Rich PIN Photodiode Using the DSB-CS Modulation Method  
Due to the optical carrier wavelength-independent property when applying the DSB-CS modulation method, microwave carrier power generation with the optical signal illuminated on one single PIN photodiode would only be related to the detector photoresponsivity, which is proportional to the input light wavelength. When the light is illuminated on the PIN photodiode for photo-electric energy conversion, the value of the photo-generated current \((I_{pin})\) generated from the photodiode is expressed as Equation (6). \[I_{pin}=RP_{MZMout} \tag{6}\]  
where \(R\) represents the photoresponsivity of the photodiode. The photoresponsivity \((R)\) of the photodiode is expressed in Equation (7). \[R=\frac{\eta e\lambda}{hc} \tag{7}\]  
where \(\eta\) is the external quantum efficiency of the photodiode. \(\lambda\) denotes the wavelength of the optical carrier. \(e=1.6\times 10^{-19}\) C, \(h=6.625\times 10^{-34}\) J - s, and \(c=3\times 10^{8}\) m/s. It can be seen that the photoresponsivity is proportional to the wavelength of the input light, demonstrating a higher photoresponsivity at longer wavelengths. Ignoring the DC components and the AC component that exceed the cut-off response frequency of the photodetector, Equation (6) can be expressed as Equation (8) based on Equations (5) and (7). \[I_{pin}\propto\left(\frac{\eta e\lambda}{hc}\right)\cdot 2E_{0}^{2}J_{1}(m)J_{- 1}(m)\cos(j2\omega_{RF}t+j2\varphi) \tag{8}\]  
Therefore, the microwave carrier power \((P_{EA})\) converted from the input light signal through the detector and measured by the electric spectrum analyzer can be expressed as Equation (9). \[P_{EA}\propto\left(I_{pin}\right)^{2}\cdot Z_{L}\propto R^{2}\cdot P_{MZMout} ^{2}\cdot Z_{L}\propto\left(\frac{\eta e\lambda}{hc}\right)^{2}\cdot P_{MZMout }^{2}\cdot Z_{L} \tag{9}\]  
where \(Z_{L}\) is the input impedance of the electric spectrum analyzer, which is generally a constant. Referring to different input wavelengths such as \(\lambda_{1}\) and \(\lambda_{2}\), Equation (10) shows the ratio between the power of the microwave carriers generated after the photodetector at wavelengths \(\lambda_{1}\) and \(\lambda_{2}\) based on Equation (9).