## 4 Phase Imbalance  
In this section, we discuss the phase imbalance parameter that may affect the performance of optical mm-wave. This comes from the influence of phase imbalance on the insertion loss as well as the DE-MZM modulation indexes. To prove that, we assumed that the inputs of upper and lower DE-MZM electrodes were imbalanced. This imbalance comes from the difference in length \(\Delta L\) between the two branches. This will cause a phase imbalance (\(\phi\)) at the MZM scheme because of different speeds of beams inside the MZM, therefore different outputs at the MZM. The phase imbalance can be represented by [28]:  
\[\phi=2\ \pi\ n\ \Delta L/\lambda \tag{3}\]  
where \(n\) is the refractive index and \(\lambda\) is the light wave wavelength. The change in \(\phi\) and splitting ratio (\(\gamma\)) of the two electrodes (upper and lower DE-MZM electrodes) may cause declines in the generated optical mm-wave. Considering the limitation or changing of the \(\gamma\) and \(\phi\) in both electrodes, the mm-wave generated can be represented by:  
Figure 4: Bessel functions of the first kind. \[ \begin{array}{ll}E(t)=\ \alpha E\ e^{i\omega_{c}t}&\left\{\gamma\Big{[}\gamma_{1} \exp\Big{(}j\pi\frac{V_{rf}\cos(\omega_{m}\ t)}{V_{\pi n}}\Big{)}+(1-\gamma_{1}) \exp\Bigl{(}-j\pi\frac{V_{rf}\cos(\omega_{m}\ t)}{V_{\pi n}}\Big{)}\Big{]}\ e^{ i\varphi}+(1-\gamma)\\ &\times\Big{[}\gamma_{2}\exp\Bigl{(}j\pi\frac{V_{rf}\cos(\omega_{m}\ t+\frac{ \gamma}{2})}{V_{\pi n}}\Bigr{)}\Big{]}\Big{]}\\ &+(1-\gamma_{2})\exp\Bigl{(}-j\pi\frac{V_{rf}\cos(\omega_{m}\ t+\frac{\gamma}{ 2})}{V_{\pi n}}\Bigr{)}\Big{]}\Big{\}}\\ &=\alpha\ E\ e^{i\omega_{c}t}\left\{\gamma_{1}\left\{\gamma_{1}\ e^{i\omega_{m }\omega_{m}t}+(1-\gamma_{1})e^{j\pi-i\cos\omega_{m}t}\right\}\ e^{i\varphi} \right.\\ &+(1-\gamma)\left[\gamma_{2}\ e^{-j\pi\sin\omega_{m}t}+(1-\gamma_{2})\ e^{j \pi\sin\omega_{m}t}\right.\big{]}\right\}\\ &=\alpha\ E\ e^{i\omega_{c}t}\left\{\gamma_{1}\sum\limits_{m=-\infty}^{\infty} \gamma_{n}\ m\ e^{i\omega_{m}t}+(1-\gamma_{1})\right.\\ &-\gamma_{1})\ \sum\limits_{m=-\infty}^{\infty}(-\overline{r})\ h_{n}\ (m)e^{i\omega_{m}t} \bigg{]}e^{i\varphi}+(1-\gamma)\\ &\times\Big{[}\gamma_{2}\ \sum\limits_{m=-\infty}^{\infty}(-1)^{n}\ h_{n}\ (m)e^{i\omega_{m}t}+(1-\gamma_{2})\ \sum\limits_{m=-\infty}^{\infty}\ J_{n}\ (m)e^{i\omega_{m}t} \Big{]}\bigg{\}}\\ &=\alpha\ E\ \sum\limits_{m=-\infty}^{\infty}[\gamma_{T1}e^{i\varphi}\ \overline{r}+\gamma \ (1-\gamma_{1})e^{i\varphi}(-j)^{n}+(1-\gamma)\gamma_{2}\ (-1)^{n}+(1-\gamma)(1\\ &-\gamma_{2})\ J_{n}(m)e^{i(\omega_{c}+n\omega_{m})t}\\ &=\alpha\ E\ \sum\limits_{m=-\infty}^{\infty}A_{n}J_{n}(m)e^{i(\omega_{c}+n \omega_{m})t}\end{array}  \tag{4}\]  
where \(E\) is the light wave amplitude and \(\alpha\) is the modulator insertion loss, \(\omega_{c}\) is an angular frequency, \(V_{rf}\) is the voltage of local oscillator, \(m\) is the modulation index, \(V_{\pi a}\) and \(V_{\pi a}\) are the switching voltages of the two upper and lower DE-MZM electrodes, and \(V_{a}\) and \(V_{b}\) are the two bias voltages where \(V_{a}=V_{b}\). The \(A_{n}\) can be expressed as:  
\[A_{n}=\left\{ \begin{array}{ll}\gamma\ e^{i\varphi}+(1-\gamma),&n=4m\\ -j\gamma(1-2\gamma_{1})e^{i\varphi}+(1-\gamma)(1-2\gamma_{2}),&n=4m+1\\ -j\gamma e^{i\varphi}+(1-\gamma),&n=4m+2\\ j\gamma(1-2\gamma_{1})e^{i\varphi}+(1-\gamma)(1-2\gamma_{2}),&n=4m+3\end{array} \right. \tag{5}\]  
Assuming that the two electrodes of DE-MZM have a \(\gamma\) of \(\gamma_{1}\) and \(\gamma_{2}\), in the case of \(\gamma_{1}\) = \(\gamma_{2}\) = 0.5 and \(\phi\) = 0, Equation (4) becomes Equation (6) [28]. The orders of \(n=4m+1,\ n=4m+2\) and \(n=4m+3\) can be ignored. \[ \begin{array}{ll}E(t)=\alpha\ E\ e^{i\omega_{c}t}&\sum\limits_{n=-\infty}^{ \infty}J_{4n}(m)e^{j(\omega_{c}+4m\omega_{m})t}\\ &\approx\alpha\ E\ \Bigl{\{}J_{0}(m)e^{i\omega_{c}t}+J_{4}(m)\Bigl{[}e^{j( \omega_{c}+4\omega_{m})t}+e^{j(\omega_{c}-4\omega_{m})t}\Bigr{]}\\ &+J_{8}(m)\Bigl{[}e^{j(\omega_{c}+8\omega_{m})t}+e^{j(\omega_{c}-8\omega_{m}) t}\Bigr{]}+\cdots\,\Bigr{\}}.\end{array}