## 3 Operation Principle  
### The Expression of the Output from MZMs  
The optical carrier from CW laser is expressed as \(E_{0}(t)=8E_{0}\exp(j\omega_{0}t)\), where \(8E_{0}\) and \(\omega_{0}\) are the amplitude and angular frequency of the optical carrier, respectively. The binary data signal is denoted as \(s(t)=\sum_{n}I_{n}g(t-nT)\), where \(I_{n}\in\{0,1\}\) is the binary sequence, \(g(t)\) is the code form function, and \(T\) is the code word duration.  
The gain of EG is denoted as G and set as 6. The phase modulation constant of PM is denoted as P and set as \(33.75^{\circ}\) (\(3\pi/16\)). The half-wave voltage \(V_{\pi}\) of the MZMs is set as \(4V\). The RF drive signal of the MZMs is expressed as follows:  
\[V_{i}(t)=V_{RF}\cos\left[\omega_{RF}t+\frac{3\pi}{16}s(t)+\varphi_{i}\right]+ 6s(t) \tag{1}\]  
Figure 2: Schematic diagram of the composite RF signal module, the \(\pm 8^{\text{th}}\)-order sideband generator module, and the data recovery module.  
In the back-to-back (BTB) system, after beating the \(\pm 8^{\text{th}}\)-order sidebands in the PD1, the frequency 16-tupling MMW is generated and can be expressed as follows:  
\[I(0,t)=\mu|E_{out}(0,t)|^{2}=2\mu E_{0}^{2}f_{8}^{2}(m)\{1+\cos[16\omega_{RF}t+ \pi s(t)]\} \tag{7}\]  
where \(\mu\) is the responsivity of the PD1.  
In the optic fiber transmission system, the \(\pm 8^{\text{th}}\)-order sidebands signals have different group velocities caused by fiber chromatic dispersion. After they transmit the distances of \(z\), the output signal from the fiber can be expressed as follows:  
\[E_{out}(z,t)=e^{-\gamma z}E_{0}\Bigg{\{}\begin{array}{l}J_{8}(m)e^{|z_{0}t+ 8\omega_{RF}t-\beta(\omega_{0}+8\omega_{RF})z|+j\pi s(t-t^{\prime})}+\\ J_{-8}(m)e^{|z_{0}t-8\omega_{RF}t-\beta(\omega_{0}-8\omega_{RF})z|}+e^{|z( \omega_{0}t-\beta\omega_{0}z)}\end{array}\Bigg{\}} \tag{8}\]  
where \(\gamma\) is the fiber loss coefficient, \(t^{\prime}=\left(\omega_{0}+8\omega_{RF}\right)^{-1}\beta(\omega_{0}+8\omega_ {RF})z\) is the time delay of the binary code word, and \(\beta(\omega)\) is the propagation constant. Since the optical power is not very high, the nonlinearity of fiber is neglected here.  
Comparing Equation (8) with Equation (6), we can see that the spectrum does not change after it transmits through the optical fiber. Injecting the output of fiber in the PD, the output electrical current from the PD can be expressed as follows:  
\[I(z,t)=\mu|E_{out}(z,t)|^{2} \tag{9}\]  
Expanding \(\beta(\omega_{0}\pm 8\omega_{RF})\) in Equation (8) with the Taylor series, in the expanded form of \(\beta(\omega_{0}\pm 8\omega_{RF})\) the terms of \(\beta^{(n)}(\omega_{0})(n\geq 3)\) are very small. Neglecting the terms of \(\beta^{(n)}(\omega_{0})(n\geq 3)\), the \(\beta(\omega_{0}\pm 8\omega_{RF})\) can be expressed as follows:  
\[\beta(\omega_{0}\pm 8\omega_{RF})\approx\beta(\omega_{0})\pm 8\omega_{RF} \beta^{\prime}(\omega_{0})+32\omega^{2}{}_{RF}\beta^{\prime\prime}(\omega_{0}) \tag{10}\]  
Substituting Equations (8) and (10) into Equation (9), the output current from the PD1 can be expressed as follows:  
\[\begin{array}{l}I(z,t)=\mu|E_{out}(z,t)|^{2}\\ =2e^{-2\gamma z}\mu E_{0}^{2}f_{8}^{2}(m)\{1+\cos[16\omega_{RF}t+\beta(\omega _{0}+8\omega_{RF})z-\beta(\omega_{0}-8\omega_{RF})z-\pi s(t-t^{\prime})]\}\\ =2e^{-2\gamma z}\mu E_{0}^{2}f_{8}^{2}(m)\{1+\cos[16\omega_{RF}t+16\omega_{RF} \beta^{\prime}(\omega_{0})z-\pi s(t-t^{\prime})]\}\end{array} \tag{11}\]  
From Equation (11), it can be seen that the code words of the data signal do not undergo bit walk-off, except for the time delay.