## 2 Optical Field of DD-MZM Output  
### DD-MZM with Basic Configuration  
Mach-Zehnder modulator (MZM) is an external IM generally used on fiber optic links. MZM can be driven on one or both arms. MZM that is driven on both arms is known as dual-drive MZM (DD-MZM). With the configuration illustrated in Figure 2, DD-MZM can produce several modulation schemes shapes. \(E_{m}(t)\) generated by the laser diode (LD) is equally divided into both arms (upper and lower arms). The MZM upper arm is driven by the RF signal (\(X_{TX}(t)\)) and is given an additional DC voltage of \(V_{bias}\). The lower arm of the MZM is driven by the same \(X_{TX}(t)\) signal, but its phase is shifted using an electrical phase shifter (EPS) of \(\theta\) without an additional DC voltage. Assuming that the extinction ratio (ER) of DD-MZM is extremely high, and the optical signal has the same polarization at every point, the optical field of DD-MZM output \(E_{dual}(t)\) can be approximated by [30]  
\[E_{dual}(t)\approx\frac{1}{2}\sqrt{2P_{in}}e^{2\pi f_{i}t}\Bigg{\{}e^{(i\pi \frac{V_{up}(t)}{V_{m}})}+e^{(i\pi\frac{V_{down}(t)}{V_{m}})}\Bigg{\}} \tag{1}\]  
where \(V_{m}\) is the MZM switching voltage, \(P_{in}\) is the optical power output of LD, \(f_{c}\) is the continuous optical wave frequency, \(V_{up}(t)\) is the upper signal drive and \(V_{down}(t)\) is the lower signal drive. \(X_{TX}(t)\) is given by  
\[X_{TX}(t)=V_{m}\cos 2\pi f_{m}t \tag{2}\]  
where \(V_{m}\) is the RF signal amplitude, \(f_{m}\) is the RF signal frequency, \(V_{up}(t)\)) and \(V_{down}(t)\) of DD-MZM are given by  
\[V_{up}(t)=V_{m}\cos(2\pi f_{m}t)+V_{bias} \tag{3}\]  
Figure 2: Basic configuration of DD-MZM to generate various modulation schemes. \[V_{down}(t)=V_{m}\cos(2\pi f_{m}t+\theta) \tag{4}\]  
where \(\theta\) is the phase of the EPS in radians. By substituting Equations (3) and (4) into Equation (1)  
\[E_{dual}(t)\approx\frac{1}{2}\sqrt{2P_{in}}e^{2\pi f_{e}t}\left\{e^{(j\pi\frac{ V_{m}}{V_{m}}\cos\left(2\pi f_{m}t\right)+j\pi\frac{V_{bias}}{V_{m}})}+e^{(j\pi \frac{V_{m}}{V_{m}}\cos\left(2\pi f_{m}t+\theta\right))}\right\} \tag{5}\]  
is obtained. For simplification, Equation (5) can be rewritten as  
\[ \begin{split} E_{dual}(t)&=\frac{1}{2}\sqrt{2P_{ in}}e^{j2\pi f_{e}t}\left\{e^{(j\pi\cos\left(2\pi f_{m}t\right)+j\pi\gamma)}+e^{ j\pi\cos\left(2\pi f_{m}t+\theta\right)}\right\}\\ &=\frac{1}{2}\sqrt{2P_{in}}e^{2\pi f_{e}t}\left\{e^{jm\cos\left( 2\pi f_{m}t\right)}.e^{j\pi\gamma}+e^{jm\cos\left(2\pi f_{m}t+\theta\right)} \right\}\end{split}  \tag{6}\]  
where \(m=\pi\frac{V_{m}}{V_{\pi}}\) is the DD-MZM modulation index, and \(\gamma=\frac{V_{bias}}{V_{\pi}}\) is the normalized bias voltage. To simplify the analysis, Equation (6) is written as  
\[E_{dual}(t)=\frac{1}{2}\sqrt{2P_{in}}e^{j2\pi f_{e}t}\left\{A.e^{(j\pi\gamma)} +B\right\} \tag{7}\]  
where  
\[A=e^{jm\cos\left(2\pi f_{m}t\right)} \tag{8}\]  
\[B=e^{jm\cos\left(2\pi f_{m}t+\theta\right)} \tag{9}\]  
By applying Jacobi Anger expansion [31], where  
\[e^{jm\cos x}=\sum_{n=-\infty}^{\infty}j^{n}J_{n}(m)e^{jnx} \tag{10}\]  
and  
\[e^{jm\sin x}=\sum_{n=-\infty}^{\infty}J_{n}(m)e^{jnx} \tag{11}\]  
Equations (8) and (9) become  
\[A=\sum_{n=-\infty}^{\infty}j^{n}.J_{n}(m).e^{jn(2\pi f_{m}t)} \tag{12}\]  
\[B=\sum_{n=-\infty}^{\infty}j^{n}.J_{n}(m).e^{jn(2\pi f_{m}t+\theta)} \tag{13}\]  
where \(J_{n}(m)\) is the \(n\)th Bessel function of the first kind; therefore, Equation (6) can be stated as  
\[ \begin{split} E_{dual}(t)&=\frac{1}{2}\sqrt{2P_{ in}}e^{j2\pi f_{e}t}\Bigg{\{}\sum\limits_{n=-\infty}^{\infty}j^{n}.J_{n}(m).e^{ j\pi\gamma}+\sum\limits_{n=-\infty}^{\infty}j^{n}.J_{n}(m).e^{jn(2\pi f_{m}t+ \theta)}\Bigg{\}}\\ &=\frac{1}{2}\sqrt{2P_{in}}\Bigg{\{}\sum\limits_{n=-\infty}^{ \infty}j^{n}.J_{n}(m).\left(e^{jn\gamma}+e^{jn\theta}\right)e^{j2\pi(f_{e}+nf _{m})t}\Bigg{\}}\end{split}  \tag{14}\]  
From Equation (14), it can be seen that the DD-MZM output optical field consists of a carrier and sideband with infinite order.