## 2 Operation Principle
The output optical fields of the DOMZM, under a condition of small signal modulation, can be written as:  
\[E_{DOMZM}(t)=\frac{1}{2}E_{m}\exp(j\cdot 2\pi f_{m}t)\left[\exp(j\frac{\pi V(t)} {2V_{\pi}})+j\exp(-j\frac{\pi V(t)}{2V_{\pi}})\right] \tag{2}\]  
where \(E_{m}\) is the amplitude of the input optical signal of the DOMZM and \(f_{m}\) is the frequency of the output optical signal of the ML. \(V_{\pi}\) is the half-wave voltage of the DOMZM. Then, Equation (2) can be expanded with Bessel functions:  
\[E_{DOMZM}(t)\approx\frac{\sqrt{2}}{2}E_{m}\left\{ \begin{array}{l}J_{0}(\beta _{1})\exp j\left(2\pi f_{m}t+\frac{\pi}{4}\right)+J_{1}(\beta_{1})\exp j\left[ 2\pi(f_{m}+f_{0})t+\frac{\pi}{4}\right]\\ +J_{1}(\beta_{1})\exp j\left[2\pi(f_{m}-f_{0})t+\frac{\pi}{4}\right]\end{array}  \right\} \tag{3}\]  
where \(\beta_{1}=\pi V_{0}2V_{\pi}\). \(J_{0}(\beta_{1})\) is the zero-order Bessel functions of the first kind. \(J_{1}(\beta_{1})\) is the first-order Bessel functions of the first kind. According to Equation (3), \(f_{m}-f_{0}\), \(f_{m}-f_{1}\), \(f_{m}-f_{2}\), \(f_{m}-f_{3}\) and \(f_{m}-f_{4}\) are the frequencies of \(-1\) st sidebands of the modulated signals, as shown in Figure 2a. As far as is known, if the detuning frequency is written as \(\Delta f=f_{m}-f_{s}\) and the range of the optical gain spectrum is changed with \(\Delta f\), then a dual-passband MPF with a tunable center frequency can be obtained. The frequency \(f_{cav}\) of the red-shifted cavity mode of the DFB can be expressed as [14; 15]:  
\[f_{cav}=f_{s}+\frac{1}{4\pi}\alpha g(N-N_{th}) \tag{4}\]  
Figure 2: Principle of the tunable dual-passband MPF. (**a**) is the optical spectrum at point A; (**b**) is the optical spectrum at point B; (**c**) is the optical spectrum at point C; (**d**) is the electrical spectrum at point D after the signals’ beating in BPD. where \(\alpha\) is the linewidth enhancement factor, \(g\) is the linear gain coefficient, \(N\) is the carrier number in the cavity, and \(N_{\text{th}}\) is the threshold carrier number. The \(-1\) st sidebands of the modulated signals, which fall into the gain spectra with center frequencies \(f_{\textit{raw1}}\) and \(f_{\textit{raw2}}\) will be selected and amplified after optically injected DFB lasers which are shown in Figure 2b,c. A dual-passband MPF will be realized after the signals' beating in BPD, which can be seen in Figure 2d. Furthermore, the resonance frequency \(f_{\textit{res}}\) between the ML and the red-shifted cavity mode of the DFB can be expressed as:  
\[f_{\textit{res}}=f_{m}-f_{\textit{raw}} \tag{5}\]  
Combining Equations (4) and (5) with the detuning frequency yields the following equation:  
\[f_{\textit{res}}=\Delta f-\frac{1}{4\pi}\alpha g(N-N_{th}) \tag{6}\]  
Considering the principle of the MPF, the center frequencies \(f_{c}\) of the two passbands in the MPF are equal to the resonance frequencies of DFB1 and DFB2, which can be expressed as:  
\[f_{c}=\Delta f-\frac{1}{4\pi}\alpha g(N-N_{th}) \tag{7}\]  
It should be noted that \(N\) is a parameter that is not only interrelated with the detuning frequency \(\Delta f\), but is also related to the injection ratio (\(R\)), which is defined as the square root of the ratio between the power of the injected light and the power of free-running slave DFB lasers [13]. The power of the injected light is controlled by tuning the power of the optical signal output from the ML when the polarization of the injected light is fixed in the MPF link. The frequency and power of the free-running DFB laser can be controlled by tuning the temperature controller and laser diode controller of the DFB laser driver in the experiment. With the change in detuning frequency or injection ratio, the center frequency of the optical gain spectrum will be changed at the same time. So, in the proposed MPF, both the center frequencies of the two passbands can be changed by adjusting the injected ratio \(R\) and detuning frequency \(\Delta f\), individually or together, with wide tunability.