# An Ultra-Wideband Microwave Photonic Channelized Receiver with Zero-IF Architecture
## 2 Principle
We can see that, if the following equations are satisfied, five flat comb lines can be generated [19]:  
\[\frac{E_{in}(t)}{2}\sin AJ_{1}(B)=\frac{E_{in}(t)}{2}\cos AJ_{2}(B)=U \tag{7}\]  
\[\frac{E_{in}(t)}{2}\Big{[}\cos AJ_{0}(B)+\cos\Big{(}\frac{\pi V_{1b}}{2V_{\pi}} \Big{)}e^{j\frac{\pi V_{1c}}{V_{\pi}}}\Big{]}=-U \tag{8}\]  
From Equations (7) and (8), the three DC bias voltages can be expressed:  
\[V_{1a}=\frac{2V_{\pi}}{\pi}\arctan[J_{2}(B)/J_{1}(B)] \tag{9}\]  
\[E_{0}=\frac{E_{in}(t)}{2}\Big{[}\cos AJ_{0}(B)+\cos\!\Big{(}\frac{\pi V_{1b}}{2 V_{\pi}}\Big{)}e^{i\frac{\pi V_{1c}}{V_{\pi}}}\Big{]} \tag{2}\]0  
\[E_{0}=\frac{E_{in}(t)}{2}\Big{[}\cos AJ_{0}(B)+\cos\!\Big{(}\frac{\pi V_{1b}}{2 V_{\pi}}\Big{)}e^{i\frac{\pi V_{1c}}{V_{\pi}}}\Big{]} \tag{2}\]1  
The half-wave voltage of the MZM is 3.5 V and the modulation index (MI) is 0.83 at this time. From Equations (9)-(11), when the three DC bias voltages of DPMZM1 are \(V_{1a}=3.15\) V, \(V_{1b}=-6.23\) V, and \(V_{1c}=0\) V, a five flat comb lines can be generated. In DPMZM2, only the \(LO_{2}\) signal with a frequency of 10 GHz and amplitude is 1.4 V is applied to drive MZMb and a push-pull configuration of the sub-MZMs is assumed. The output field of the DPMZM2 can be expressed as:  
\[E_{0}=\frac{E_{in}(t)}{2}\Big{[}\cos AJ_{0}(B)+\cos\!\Big{(}\frac{\pi V_{1b}}{2 V_{\pi}}\Big{)}e^{i\frac{\pi V_{1c}}{V_{\pi}}}\Big{]} \tag{2}\]2  
\[E_{0}=\frac{E_{in}(t)}{2}\Big{[}\cos AJ_{0}(B)+\cos\!\Big{(}\frac{\pi V_{1b}}{2 V_{\pi}}\Big{)}e^{i\frac{\pi V_{1c}}{V_{\pi}}}\Big{]} \tag{2}\]3  
In order to output even-order optical bands, it can be operated at the maximum transmission point (MATP). Therefore, we can get \(V_{2a}=0\), where \(A=\pi V_{1a}/2V_{\pi}\), \(B=\pi V_{0}/2V_{\pi}\), and \(\omega_{LO2}=2\pi f\). The optical signal output from the MZMa can be rewritten as:  
\[E_{0}=\frac{E_{in}(t)}{2}\Big{[}\cos AJ_{0}(B)+\cos\!\Big{(}\frac{\pi V_{1b}}{2 V_{\pi}}\Big{)}e^{i\frac{\pi V_{1c}}{V_{\pi}}}\Big{]} \tag{2}\]4  
In order to suppress the carrier of MZMa to the greatest extent, MZMb is set to work at the minimum transmission point (MITP), finally, the second-order optical bands of carrier suppression are obtained after modulation. \[E_{0}=\frac{E_{in}(t)}{2}\Big{[}\cos AJ_{0}(B)+\cos\!\Big{(}\frac{\pi V_{1b}}{2 V_{\pi}}\Big{)}e^{i\frac{\pi V_{1c}}{V_{\pi}}}\Big{]} \tag{2}\]5  
\[E_{0}=\frac{E_{in}(t)}{2}\Big{[}\cos AJ_{0}(B)+\cos\!\Big{(}\frac{\pi V_{1b}}{2 V_{\pi}}\Big{)}e^{i\frac{\pi V_{1c}}{V_{\pi}}}\Big{]} \tag{2}\]6  
From Equations (14)-(16), the three DC bias voltages of DPMZM2 are \(V_{2a}=0\) V, \(V_{2b}=0.83\) V, and \(V_{2c}=0\) V. The DPMZM2 outputs positive and negative second-order optical bands with a suppressed carrier. The positive and negative second-order sideband frequencies are \(-20\) GHz and \(20\) GHz with respect to 193.1 THz, respectively. The output field of DPMZM2 can be expressed as:  
\[E_{0}=\frac{E_{in}(t)}{2}\Big{[}\cos AJ_{0}(B)+\cos\!\Big{(}\frac{\pi V_{1b}}{2 V_{\pi}}\Big{)}e^{i\frac{\pi V_{1c}}{V_{\pi}}}\Big{]} \tag{2}\]7In brief, Equations (1)-(11) explain the generation of five comb lines, and Equations (12)-(17) explain the generation of second-order optical bands of carrier suppression. The two optical signals output from DPMZM1 and DPMZM2 are combined by the following PBC, and then they travel out of the DP-QPSK modulator. The combined optical signal can be expressed as:  
\[E_{0}=\frac{E_{in}(t)}{2}\Big{[}\cos AJ_{0}(B)+\cos\!\Big{(}\frac{\pi V_{1b}}{2 V_{\pi}}\Big{)}e^{i\frac{\pi V_{1c}}{V_{\pi}}}\Big{]} \tag{2}\]8  
where \(\overset{\rightarrow}{e}_{TE}\) and \(\overset{\rightarrow}{e}_{TM}\) denote the unit vectors of _TE_ and _TM_ modes, respectively. The optical frequency comb from the DPMZM1 output is in the _TE_ mode, and the positive and negative second-order optical sidebands from the DPMZM2 output are in the _TM_ mode.