# Generation of Flat Optical Frequency Comb Using Cascaded Electro-Optic Modulators
## 2 Principles  
The cascade scheme we implemented is illustrated in Figure 1. The CW light generated by the laser diode first passes through the MZM, followed by two _P_Ms in succession. Finally, the output spectrum is observed with an Optical Spectrum Analyzer (OSA) at the end of the setup. We assume that the input signal \(E_{in}(t)\) generated by the laser is  
\[E_{in}(t)=E_{0}\exp(j\omega_{c}t)\]\(E_{0}\) indicates the electric field strength of the input signal. \(\omega_{c}\) indicates the input signal frequency. When the input optical signal passes through the MZM, the expression of the optical signal is as follows:  
\[E_{out1}(t)=\frac{E_{in}(t)}{2}\left\{\begin{array}{c}\exp(j\Delta\varphi) \biggl{[}\sum\limits_{n=-\infty}^{\infty}J_{n}(A_{1})\exp(jn\omega_{RF}t) \biggr{]}\\ +\biggl{[}\sum\limits_{n=-\infty}^{\infty}J_{n}(A_{2})\exp(jn\omega_{RF}t+jn \theta)\biggr{]}\end{array}\right\}\]  
where \(\omega_{RF}\) represents the \(RF\) signal frequency, \(\Delta\varphi=\varphi_{2}-\varphi_{1}=\pi V_{bias1}/V_{\pi-DC1}-\pi V_{bias2}/V_ {\pi-DC2}\) represents the direct current (\(DC\)) phase difference between the two arms of the MZM (the static phase shift), \(A_{1}=\pi V_{RF1}/V_{\pi 1}\) represents the modulation index of the upper arm of the MZM, \(A_{2}=\pi V_{RF2}/V_{\pi 2}\) represents the modulation index of the lower arm of the MZM, and \(\theta\) represents the phase difference in \(RF\) signal between the upper and lower arms. \(V_{RF1}\) and \(V_{RF2}\) represent the \(RF\) signal amplitudes of the upper and lower arms. \(V_{\pi 1}\) and \(V_{\pi 2}\) represent the half-wave voltages of the upper and lower arms. \(V_{bias1}\) and \(V_{bias2}\) indicate the bias voltage amplitudes for the upper and lower arms. Additionally, \(V_{\pi-DC1}\) and \(V_{\pi-DC2}\) represent the \(DC\) half-wave voltage of the upper and lower arms of the MZM. After the light signal passes through two _PMs_, the final output signal can be expressed as follows:  
\[E_{out2}(t)=\frac{E_{in}(t)}{2}\sum\limits_{l}\left\{\sum\limits_{k}J_{l-k}(A _{3})\biggl{[}J_{k}(A_{1})e^{i\Delta\varphi}+J_{k}(A_{2})e^{i\theta}\biggr{]} \right\}e^{il\omega_{RF}t}\]  
The two _PMs_ used in our scheme are identical, and thus, we treated the two _PMs_ in the subsequent cascade as equivalent components, which are represented by the modulation index \(A_{3}\). Since the modulator's performance remains consistent, applying a voltage to the two _PMs_ separately can be considered equivalent to applying a voltage to a single _PM_. However, the final modulation index must be divided by two to correspond to each _PM_. The equation for the modulation index is given by  
\[A_{3}=2\pi V_{RF-PM}/V_{\pi-PM}\]  
In this equation, \(V_{RF-PM}\) represents the amplitude of the \(RF\) signal applied to the _PM_, while \(V_{\pi-PM}\) denotes the half-wave voltage of the _PM_. The expression above clearly shows that the resulting output optical signal consists of a series of harmonics when the CW light passes through the cascaded modulators. The amplitude of each harmonic is associated with the corresponding first-class Bessel function (which depends on the modulation index) and the magnitude of the modulation signal phase.  
Figure 1: Scheme of OFC generated by cascaded electro-optic modulators. PS: phase shifter. \(DC\): direct current. \(RF\): radio frequency.