## 2 Principle and Simulation
For DDMZM2, it is set at the maximum transmission point (MATP), where \(\Delta\phi_{34}=0\). When the condition above is inserted in Equation (4), the optical signal output by the Dpol-DDMZM can be written as follows:  
\[ \begin{array}{rl}E_{\text{out}}(t)&=E_{\text{out}}(t)\\ &=\frac{E_{\text{in}}(t)}{2}\end{array} \Big{\{} \begin{array}{rl}\exp(j\phi_ {2})\Big{[}e^{j\pi/2+j\beta\cos(\omega_{RF}t)}+e^{-j\beta\cos(\omega_{RF}t)} \Big{]}\\ &+\exp(j\phi_{4})\Big{[}e^{j\beta\cos(\omega_{RF}t+\delta)}+e^{-j\beta\cos( \omega_{RF}t+\delta)}\Big{]}\end{array} \Big{\}} \tag{6}\]  
Because two DDMZMs work in quadrature polarization directions, the optical signal will not interfere with each other. When this optical signal is sent to a PD for intensity detection, the output current of the PD is given by the following equation:  
\[ \begin{array}{rl}I(t)&=I_{1}(t)+I_{2}(t)\\ &=\Re|E_{in}|^{2}\{I_{DC}-[J_{1}(2\beta)\sin(\Omega t+\pi/2)+J_{3}(2\beta)\sin (3\Omega t+3\pi/2)+J_{2}(2\beta)\ \cos(2\Omega t+2\delta)]\}\end{array}  \tag{7}\]  
where \(\Re\) is the responsivity of PD, \(I_{DC}\) represents \(1+J_{0}(2\beta)/2\), \(J_{n}(2\beta)\) denotes the n-th order of the first kind of the Bessel function. By comparing Equation (7) with Equation (5), we can see that if the following conditions can be satisfied, then a sawtooth waveform can be generated:  
\[\left\{ \begin{array}{rl}\delta=\pi/4\\ r_{1/2}=J_{1}(2\beta)/J_{2}(2\beta)=2\\ r_{1/3}=J_{1}(2\beta)/J_{3}(2\beta)=3\end{array} \right. \tag{8}\]  
The phase difference, \(\delta\), can be adjusted by varying the PS. In addition, the ratio between the Bessel functions can be adjusted by changing the modulation depth of each DDMZM. The relationship of the Bessel functions and the modulation index are plotted in Figure 2. As we can see, when \(2\beta=1.73\), the ratio \(r_{1/2}=2\) is satisfied. Moreover, if we set the modulation index of DDMZM2 to \(2\beta=2.30\), then \(r_{1/3}=3\) is satisfied. \[ \begin{array}{rl}I(t)=\Re|E_{in}|^{2}\{I_{DC}-J_{1}(2\beta)\left[\sin( \Omega t+\frac{\pi}{2})+\frac{1}{2}\sin(2\Omega t+\pi)+\frac{1}{3}\sin(3\Omega t +\frac{3\pi}{2})\right]\}\end{array}  \tag{9}\]  
Comparing Equation (9) with Equation (5), the relationship between frequency and phase satisfies the requirement for sawtooth waveform generation, and the desired full-duty sawtooth pulse train is successfully obtained. 2. Triangular and Square waveform  
Similarly, if both DDMZMs are operated at QATP, that is, \(\Delta\phi_{12}=\Delta\phi_{34}=\pi/2\), the optical signal output by the Dpol-DDMZM can be written as follows:  
\[ \begin{array}{ll}E_{\text{out}}(t)&=E_{\text{out}1}(t)&+E_{\text{out}2}(t)\\ &=\frac{E_{in}(t)}{2}&\Big{\{}\exp(j\phi_{2})\Big{[}e^{j\pi/2+j\beta\cos(\omega _{RF}t)}+e^{-j\beta\cos(\omega_{RF}t)}\Big{]}\\ &&+\exp(j\phi_{4})\Big{[}e^{j\pi/2+j\beta\cos(\omega_{RF}t+\delta)}+e^{-j\beta \cos(\omega_{RF}t+\delta)}\Big{]}\ \Big{\}}\end{array}  \tag{10}\]  
The current output after the PD can be expressed as the following equation:  
\[ \begin{array}{ll}I(t)&=I_{1}(t)+I_{2}(t)\\ &=\frac{\Re}{2}|E_{in}|^{2}\{1+2J_{1}(2\beta)\,\sin(\Omega t+\frac{\theta}{2} )+2J_{3}(2\beta)\sin(3\Omega t+\frac{3\theta}{2})\}\end{array}  \tag{11}\]  
We adjust the modulation depth of two DDMZMs to satisfy the following Bessel function: \(r_{1/3}=J_{1}(2\beta)/J_{3}(2\beta)=9\). According to Figure 2, the modulation index is \(2\beta=1.50\). Additionally, by setting \(\delta=\pi/2\), Equation (11) can be simplified as follows:  
\[ \begin{array}{ll}I(t)&=\frac{\Re}{2}|E_{in}|^{2}J_{1}(2\beta)\{I_{DC}+\cos( \Omega t)+\frac{1}{9}\cos(3\Omega t+\pi)+\cos(\Omega t+\frac{\pi}{2})+\frac{1 }{9}\cos(3\Omega t+\frac{5}{2}\pi)\}\\ &=\frac{\Re}{2}|E_{in}|^{2}J_{1}(2\beta)\{I_{DC}+\sqrt{2}\cos(\Omega t+\frac{ \pi}{4})+\frac{\sqrt{2}}{9}\cos(3\Omega t+\frac{3}{4}\pi)\}\end{array}  \tag{12}\]  
where \(I_{DC}\) represents the DC current.