## 2 Basic Theory
### The Double Sideband with Carrier Suppression (DSB-CS) Modulation Method
\[\frac{P_{EA_{1}}}{P_{EA_{2}}}=\frac{R_{\lambda_{1}}^{2}\cdot P_{\lambda_{1}MZMout }^{2}\cdot Z_{L}}{R_{\lambda_{2}}^{2}\cdot P_{\lambda_{2}MZMout}^{2}\cdot Z_{ L}}=\frac{\left(\eta_{\lambda_{1}}\lambda_{1}\right)^{2}\cdot P_{\lambda_{1}MZMout}^{2}}{ \left(\eta_{\lambda_{2}}\lambda_{2}\right)^{2}\cdot P_{\lambda_{2}MZMout}^{2}} \tag{10}\]  
Here, the values of \(P_{\lambda_{1}\ MZMout}\) and \(P_{\lambda_{2}\ MZMout}\) are set as the same for comparing the electrical power generation efficiencies at different wavelengths based on the same photodiode to demonstrate the relationship between the photo-generated microwave carrier power and the incident light wavelength of the photodiode. Then, Equation (10) can be expressed as Equation (11). \[\frac{P_{EA_{1}}}{P_{EA_{2}}}=\frac{\left(\eta_{\lambda_{1}}\lambda_{1}\right) ^{2}\cdot P_{\lambda_{1}MZMout}^{2}}{\left(\eta_{\lambda_{2}}\lambda_{2} \right)^{2}\cdot P_{\lambda_{2}MZMout}^{2}}=\frac{\left(\eta_{\lambda_{1}} \lambda_{1}\right)^{2}}{\left(\eta_{\lambda_{2}}\lambda_{2}\right)^{2}} \tag{11}\]Under the ideal circumstance, the external quantum efficiency \(\left(\eta_{\lambda_{1}}\right)\) and \(\left(\eta_{\lambda_{2}}\right)\) are the same. Therefore, the ratio is shown as Equation (12). \[\frac{P_{E\lambda_{1}}}{P_{E\lambda_{2}}}=\frac{\lambda_{1}^{2}}{\lambda_{2}^{2}} \tag{12}\]  
It is shown in Equation (12) that, if \(\lambda_{1}\) is longer than the \(\lambda_{2}\), \(P_{E\lambda_{1}}\) is larger than \(P_{E\lambda_{2}}\) demonstrating a higher microwave carrier power obtained at the longer wavelengths. Actually, for a real detector, \(\eta_{\lambda_{1}}\
eq\eta_{\lambda_{2}}\). To exclude the affection of such a case for obtaining the opto-electric power conversion efficiency, the microwave carrier power is normalized to the square of the referring external quantum efficiency. Therefore, we have the function in Equation (13). \[\left(\frac{P_{E\lambda_{1}}}{\eta_{\lambda_{1}}^{2}}\right)\Big{/}\Bigg{(} \frac{P_{E\lambda_{2}}}{\eta_{\lambda_{2}}^{2}}\Bigg{)}=\frac{\lambda_{1}^{2}} {\lambda_{2}^{2}}=r_{n(\lambda_{1}/\lambda_{2})} \tag{13}\]  
where \(\left(P_{E\lambda_{1}\;\sigma\;2}\right)/\left(\eta_{\lambda_{1}\;\sigma\;2}^{ 2}\right)\) represents the microwave carrier power per squared external quantum efficiency referring to the opto-electric power conversion efficiency under the same input optical signal power. \(r_{n(\lambda_{1}/\lambda_{2})}\) represents the ratio between the normalized microwave carrier power at the wavelengths of \(\lambda_{1}\) and \(\lambda_{2}\), indicating the variation of the opto-electric power conversion efficiency due to the difference of the input wavelengths. It is obviously shown from Equation (13) that \(r_{n(\lambda_{1}/\lambda_{2})}>1\) when \(\lambda_{1}>\lambda_{2}\). This indicates a higher microwave carrier power per squared external quantum efficiency corresponding to a larger opto-electric power conversion efficiency obtained at the longer wavelengths. Based on the deduction of the abovementioned equations, it is theoretically demonstrated that a higher photo-to-electric power conversion efficiency can be achieved at longer wavelengths under the same optical signal power benefitting from the higher photosepusivity of the detector at longer wavelengths.