## 2 Architecture of the Proposed Optical Metro Access Network
### The Principal of the OMC Generator  
The OMC generator is based on the cascaded configuration of a single phase modulator (PM) and two Mach Zehnder modulators (MZMs). This proposed technique uses electro-optic modulators driven by an RF source that produces a multiharmonic signal, separated by a fixed frequency termed frequency spacing that can be modified using the RF signal. The optical input of the PM is connected to the CW laser source, whereby the first MZM (MZM1)'s optical input is driven by the PM's output and the second MZM (MZM2) by the output of the first MZM. The phase modulator (PM) changes the injected optical signal in response to the provided amplified RF signal. Consider the optical continuous wave (CW) with electric field \(E_{0}\) and angular frequency \(\omega_{0}\). The CW's signal before entering the PM can be given as \(E_{cw}(t)=E_{0}e^{i\omega_{0}t}\), where \(E_{0}\) represents the amplitude, and \(\omega_{0}\) represents the angular frequency of the laser. The RF amplified sinusoidal signal can be described as \(V_{RF}(t)=V_{\max}(\cos(\omega_{RF}t+\phi)\), where \(V_{max}\) and \(\omega_{RF}\) represent the amplitude and angular frequency of the RF source. The phase modulator produces a phase shift that is proportional to the provided RF signal, which can be given as \(\phi\). Therefore, the phase-modulated optical signal can be given as  
\[E_{PM}(t)=E_{cw}(t)\exp(jG\beta\sin(\omega_{RF}t+\phi)) \tag{1}\]  
where \(G\) is the gain in the RF signal, and we will consider it a constant value \(\sim\)1 for ease of understanding the further equations, \(\beta\) is the modulation index of the PM, \(\phi\) is the phase of the RF signal, and \(t\) represents the time. Therefore, the term \(\beta\sin(\omega_{RF}+\phi)\) represents the phase modulation introduced by the PM. After expanding the above equation, it can be given as  
\[E_{PM}(t)=E_{0}\exp(j\omega_{0}t)\exp(j\beta\sin(\omega_{RF}t+\phi)) \tag{2}\]  
Equation (2) can be expanded further by using the well-known Jacobi-Anger expansion to express the phase-modulated signal in terms of the Bessel functions, as given below:  
\[E_{PM}(t)=E_{0}\exp(j\omega_{0}t)\sum_{n=-\infty}^{n=-\infty}J_{n}(\beta)\exp (jn(\omega_{RF}t+\phi)) \tag{3}\]  
where \(J_{n}\) is the \(n\)-th order of the Bessel function of the first kind. This expression shows the creation of sidebands with frequencies \(\omega_{0}\pm n\omega_{RF}\), with their amplitude being determined by the Bessel function \(J_{n}(\beta)\), and \(\omega_{RF}\) represents the frequency spacing between the comb lines. This equation captures the essence of phase modulation under the impact of an RF signal. The Bessel functions represent the energy distribution created due to modulation among various sidebands. The output of the PM is shown in Figure 3a. The output of the PM is connected in cascade to the first MZM. It modulates the intensity of the optical signal whose output can be given as  
\[ \begin{array}{ll}E_{MZMI}(t)&=E_{PM}(t).[1+m\cos(\omega_{RF}t)]\\ &=E_{0}\exp(j\omega_{0}t)\sum_{n=-\infty}^{n=\infty}J_{n}(\beta)\exp(jn( \omega_{RF}t+\phi)).[1+m\cos(\omega_{RF}t)]\end{array}  \tag{4}\]As \(\cos(\omega_{RF}t)=\frac{1}{2}[\exp(j\omega_{RF}t)+\exp(-j\omega_{RF}t)]\), therefore, after putting this in the above Equation (4), it becomes  
\[ \begin{array}{c}E_{MZM1}(t)=E_{0}\exp(j\omega_{0}t)\sum\limits_{n=-\infty}^{n =\infty}J_{n}(\beta)\exp(jn(\omega_{RF}t+\phi)).\\ \big{[}1+\frac{m}{2}\{\exp(j\omega_{RF}t)+\exp(-j\omega_{RF}t)\}\big{]}\end{array}  \tag{5}\]  
Here, \(m\) shows the modulation depth of the MZM that controls the additional sidebands generated by the MZM. The above expression shows a sum of frequency components that is generated by the PM and then further modulated by the first MZM, where every term corresponds to a separate frequency tone at both sides of the central frequency -\(j_{0}(\beta)\). This comb is practically finite, because the Bessel function decreases with increasing of the \"\(n\)\".