## 3 Modulation Index  
The DSB-OCS system has been modelled to investigate the RoF system performance and the signal power of the mm-wave at various modulation indexes (\(m\)). At the DSB-OCS  
Figure 1: DSB-OCS schematic diagram.  
Figure 2: DSB-OCS simulation setup.  
simulation setup, the parameters were kept at constant values and the fiber length was set to 20 km. The important findings were collected and are plotted in Figures 3 and 4. The result in Figure 3 shows the relationship between the mm-wave signal power and the modulation index. In this figure, the amplitude of the LO signals ranges from 0 V to 20 V, and the modulation index ranges from 0 to 4. It can be seen that all curves perform the same behavior in the graph with small differences of around 0.6%, starting at a modulation index of 2.6. However, the 10 GHz and 20 GHz curves have the same output for the whole iteration.  
Initially, 10.7 dBm signal power of the mm-wave was generated at a modulation index of 0. When the modulation index as raised, the curves increased gradually to reach the maximum value at 11.8 dBm for modulation indexes from 0.8 to 1. This case can be clarified by Equation (1):  
\[E_{out}=-E_{o}\cos(\Omega\mathrm{t})\ \sum_{n=0}^{\infty}\ (-1)^{n}\ J_{2n+1} \Big{(}\frac{\pi m}{2}\Big{)}\cdot\ \cos[(2n+1)\omega_{RF}t] \tag{1}\]  
where \(J_{2\pi+1}\left(\pi m/2\right)\) refers to Bessel functions of the first kind of \(n\) order where \(n=0,\ 1,\ 2,\ 3\) and the related functions \(J_{1}\,\ J_{3}\) and \(J_{5}\) can be represented by Equation (2):  
\[J_{n}\ (x)=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}\ \big{(}\frac{x}{2}\big{)}^{n+2k}}{k!\ \Gamma\ (n+k+1)} \tag{2}\]  
where \(x=\pi m/2\), \(m\) is the modulation index, or \(V_{DC}\)/\(V_{\pi}\) of the DSB-OCS, \(V_{DC}\) is the DC bias voltage, \(n\) and \(k\) are integers, and \(\omega_{RF}\) is the optical carrier frequency. Figure 4 displays the theoretical Bessel function with \(J_{1}\,\ J_{3}\) and \(J_{5}\) for comparison. Due to low bias applied, only odd Bessel components are considered. Theoretically, the \(J_{1}\) portion is dominant compared to \(J_{3}\) and \(J_{5}\) at modulation indexes of 0.6 and 1.1, while \(J_{0}\) and other components are suppressed in the DSB-OCS system. \(J_{1}\) is considered as the dual sideband components, while the dual sideband components for \(J_{3}\) and \(J_{5}\) can be regarded as noise. Theoretically, the term \(J_{1}-J_{3}-J_{5}\) is the output result of the system. It can be seen that the theoretical \(J_{1}-J_{3}-J_{5}\) term, or the output result of the system, shows maximum mm-wave signal power at a modulation index of 0.6 to 1.1 in Figure 4, and this result coincides with the simulation result in Figure 3.  
Figure 3: Signal power of optical mm-wave as a function of modulation index (\(m\)) for different LO frequencies.  
In Figure 3, the curve of 20 GHz LO records the minimum amplitude values when modulation indexes range from 2.4 to 2.8, while the curves for 10 GHz and 30 GHz record the minimum amplitude values at a modulation index ranging from 2.6 to 3. The 20 GHz curve had higher amplitude after a modulation index of 3 than the curves of 10 GHz and 30 GHz. All curves increased gradually after a modulation index of 3, with a difference of 0.65 dB in the amplitude between the 20 GHz curve and 10 GHz curve. This can also be explained by referring to the \(J_{1}-J_{3}-J_{5}\) term plot in Figure 4, where the results coincide with results in Figure 3, and it has been shown that in order to obtain the maximum mm-wave signal power amplitude, the modulation index should be between 0.6 and 1.1.