## 2 Principle and Simulation  
The configuration of the proposed waveform generation scheme is shown in Figure 1. In the proposed scheme, an optical carrier generated from an LD is coupled into a pol-DDMZM with two-quadrature polarizer DDMZMs. Two sub-MZMs (DDMZM1 and DDMZM2) are embedded in each arm of the main modulator. Each of the DDMZMs has two RF input ports and two DC input biases, respectively. The sinusoidal RF signal is split into two channels by the 3 dB divider, with one output port connected to DDMZM1 and the other output port connected to DDMZM2 via the PS. The optical signal is modulated by the RF signal via the Dpol-DDMZM to generate optical sidebands. Assume that the input optical carrier in LD is expressed as \(E_{in}(t)=E_{in}\)\(\exp(j\omega_{0}t)\), \(V_{RF}\) and \(W_{RF}\) are the amplitude and frequency of the RF signal, and \(V_{DC1}\) and \(V_{DC2}\) denote the two bias voltages applied to the upper and lower arms of the DDMZM, respectively. From [18], the optical field at the output of DDMZM can be expressed as follows:  
\[E_{\text{out}}(t)=\frac{E_{in}(t)}{\sqrt{2}}\exp(j\phi_{2})\Big{[}e^{j\Delta \phi+j\beta\cos(\omega_{RF}t)}+e^{j\beta\cos(\omega_{RF}t+\theta)}\Big{]} \tag{1}\]  
where \(\phi_{1}=\pi V_{DC1}/V_{\pi}\) and \(\phi_{2}=\pi V_{DC2}/V_{\pi}\) are phase shifts determined by the bias voltages, \(\Delta\phi=|\phi_{1}-\phi_{2}|=|\pi(V_{DC1}-V_{DC2})/V_{\pi}|\) represents the phase difference caused by the two bias voltages, and \(\beta=\pi V_{RF}/2V_{\pi}\) is the modulation index of the DDMZM; \(\theta\) is the phase difference of the RF signals applied to the two arms, where \(\theta=\pi\), and the DDMZM works in a complementary push-pull state. In this scheme, the optical field at the output of two sub-MZMs (DDMZM1 and DDMZM2) can be respectively expressed as follows:  
\[E_{\text{out}}(t)=\frac{E_{in}(t)}{2}\exp(j\phi_{2})\Big{[}e^{j\Delta\phi_{12} +j\beta\cos(\omega_{RF}t)}+e^{-j\beta\cos(\omega_{RF}t)}\Big{]} \tag{2}\]  
where \(\Delta\phi_{12}=|\phi_{1}-\phi_{2}|=|\pi(V_{DC1}-V_{DC2})/V_{\pi}|\) represents the phase difference of the DDMZM1 caused by the two bias voltages, \(\Delta\phi_{34}=|\phi_{3}-\phi_{4}|=|\pi(V_{DC3}-V_{DC4})/V_{\pi}|\) represents the phase difference of the DDMZM2 caused by the two bias voltages, \(\delta\) represents the phase difference of the RF signal loaded by the upper DDMZM1 and the lower DDMZM2 of the Dpol-DDMZM. \[E_{\text{out}2}(t)=\frac{E_{in}(t)}{2}\exp(j\phi_{4})\Big{[}e^{j\Delta\phi_{3 4}+j\beta\cos(\omega_{RF}t+\delta)}+e^{-j\beta\cos(\omega_{RF}t+\delta)}\Big{]} \tag{3}\]  
The optical field at the output of Dpol-DDMZM can be expressed as follows:  
\[ \begin{array}{llll}E_{\text{out}}(t)&=E_{\text{out}1}(t)&+E_{\text{out}2}(t )\\ &=\frac{E_{in}(t)}{2}&\Big{\{}\exp(j\phi_{2})\Big{[}e^{j\Delta\phi_{12}+j\beta \cos(\omega_{RF}t)}+e^{-j\beta\cos(\omega_{RF}t)}\Big{]}\\ &&+\exp(j\phi_{4})\Big{[}e^{j\Delta\phi_{34}+j\beta\cos(\omega_{RF}t+\delta)}+ e^{-j\beta\cos(\omega_{RF}t+\delta)}\Big{]}\end{array}  \tag{4}\]  
Figure 1: Schematic diagram of arbitrary waveform generation based on Dpol-DDMZM. PS: Phase Shifter; PR: Polarization Rotator; PBC: Polarization Beam Combiner; OSC: Oscilloscope; ESA: Electrical spectrum analyzer. The Fourier expansion of the sawtooth, triangular, and square waveforms can be approximately expressed as follows:  
\[\left\{ \begin{array}{rl}f_{saw}(t)=DC_{saw}+A_{saw}\sum\limits_{n=1}^{\infty} \frac{\sin(n\omega t)}{n}\approx DC_{saw}+A_{saw}\Big{[}\sin(\omega t)+\frac{ \sin(2\omega t)}{2}+\frac{\sin(3\omega t)}{3}+\ldots\Big{]}\\ f_{tr}(t)=DC_{tr}+A_{tr}\sum\limits_{n=1,3,5,\ldots}\frac{1}{n^{2}}\cos(n \omega t)\approx DC_{tr}+A_{tr}\Big{[}\cos(\omega t)+\frac{1}{9}\cos(3\omega t )+\ldots\Big{]}\\ f_{sq}(t)=DC_{sq}+A_{sq}\sum\limits_{n=1,3,5,\ldots}\frac{1}{n}\sin(n\omega t) \approx DC_{sq}+A_{sq}\Big{[}\sin(\omega t)+\frac{1}{3}\sin(3\omega t)+\ldots \Big{]}\end{array} \right. \tag{5}\]  
Sawtooth waveform  
If we set \(\Delta\phi_{12}=\pi/2\) in DDMZM1, it is operated at the quadrature transmission point (QATP).