# Effect of Phase Noise on the Optical Millimeter-Wave Signal in the DWDM-RoF System
## 2 Phase Noise and Phase Imbalance Effects  
The MZM scheme is still affected by the uncontrolled phase in the optical part, which creates an amplitude error in the electrical and wireless signal. Using many MZMs in the system increases the phase imbalance, so by using a single modulator, the phase imbalance effects are limited and controlled. The phase imbalance parameter influences the performance of the optical mm wave, as discussed in our prior work [18]. This is due to the difference in length \(\Delta\)L between the two branches of the MZM, which results in different beam speeds inside the MZM. These different speeds create a difference between phase \(\phi_{1}\) and \(\phi_{2}\) as shown in Figure 1. The phase imbalance can be represented by:  
\[\Delta\varphi=2\pi n\Delta\mathrm{L}/\lambda \tag{1}\]  
where \(n\) is the refractive index, and \(\lambda\) is the light wave wavelength. In our study, the phase imbalance was controlled by adjusting the phase at the phase shift. The proposed schematic diagram of the MZM for the DWDM-RoF system was driven by three input sources, as shown in Figure 2. The three input sources were the continuous wave (CW) laser, data consisting of a nonreturn-to-zero (NRZ) generated from a pseudorandom binary sequence (PRBS), and the radio-frequency (RF) signal. Therefore, the output signal \(\mathrm{E}_{\mathrm{MZM}}\) of the MZM was a combination of the three. The following equations expressed the three sources of MZM. The first source is the pseudorandom binary sequence (PRBS) data. It can be represented by:  
\[\mathrm{T}_{1}\left(\mathrm{t}\right)=\sin\omega_{\mathrm{m}}\pi^{2}(\mathrm{ t}) \tag{2}\]  
Figure 1: The phase imbalance in MZM: (**a**) MZM diagram and (**b**) phase difference in the mm-wave signal. Figure 2: The schematic diagram of an MZM. where \(\mathrm{T_{1}}\) (t) is the electric field of the transmitted data represented by a sine wave of the signal and \(\omega_{\mathrm{m}}\) is the angular frequency (\(\omega_{\mathrm{m}}=2\pi\mathrm{f_{m}}\)). The second source was the continuous wave (CW) laser. It can be represented by:  
\[\mathrm{T_{2}}\ (\mathrm{t})=\mathrm{e}^{j\omega_{\mathrm{c}}\mathrm{t}} \tag{3}\]  
where \(\mathrm{T_{2}}\ (\mathrm{t})\) is the optical field of the optical carrier, \(j\) is the Bessel function of the first kind, and \(\omega_{\mathrm{c}}\) is the light-wave angular frequency (\(\omega_{\mathrm{c}}=2\pi\mathrm{c}/\lambda\)) in scalar form (scalar form is used based on the assumption that the light is linearly polarized, and it is aligned with the corresponding polarization state axis of the modulator) [21]. The third source was the radio frequency (RF) signal. It can be represented by:  
\[\mathrm{T_{3}}\ (\mathrm{t})=\sin\omega_{\mathrm{m}}t+\varphi \tag{4}\]  
where \(\mathrm{T_{3}}\ (\mathrm{t})\) is the electrical field of the RF signal and \(\varphi\) is the initial phase of electrical drive signal. This phase could be adjusted to control the phase imbalance that occurred at the MZM. According to Equations (2)-(4), the output signal \(\mathrm{E_{MZM}}\) of the MZM can be expressed by:  
\[\mathrm{E_{MZM}}=\ \alpha\sum\sin\omega_{\mathrm{m}}\pi^{2}(\mathrm{t})+ \mathrm{e}^{j\omega_{\mathrm{c}}\mathrm{t}}+\sin\omega_{\mathrm{m}}\mathrm{t}+\varphi \tag{5}\]  
where \(\alpha\) is the insertion losses of the modulator, and \(\alpha\) is significantly affecting the signal power, as reported [17; 18; 19]. The phase shift (PS) was used as a controller after the RF signal to control the phase imbalance that occurred at the MZM. The PS can adjust the phase imbalance that affects the mm-wave signal. The phase imbalance was initially calculated at the output of the MZM, and then the phase was adjusted at the PS to decrease the phase imbalance.