## 3 Frequency Response  
The frequency response was measured using a vector network analyzer (VNA, Anritsu 37269E-R, Atsugi, Japan). Small sinusoidal signals from the VNA were applied to the RF electrode of a single MZ arm through an RF probe (typical insertion loss is 0.8 dB), and the modulated signals were then returned to the VNA through an erbium-doped fiber amplifier (EDFA, Alnair Labs CPA-100-CL, Shinagawa, Japan), a band-pass filter (BPF, Alnair Labs CVF-220-CL, Shinagawa, Japan) and an opto-electronic (OE) converter (Anritsu MN4765, Atsugi, Japan). Figure 3 shows \(S_{11}\) for the RF signals reflected to the output port of the VNA and \(S_{21}\) for the modulated optical signals that were input into the VNA. As previously mentioned, the device did not have termination resistors, and thus \(S_{11}\) becomes 0 dB at low frequencies. At high frequencies, on the other hand, the RF signals tend to attenuate during the transmission due to the parasitic resistance at the p-n junction, which decreases \(S_{11}\) gradually. The dip appearing at 19 GHz is considered to be caused by the excitation of the coupled slotline mode. This will be suppressed by symmetrizing the GSG RF electrode, which was designed asymmetrically in this fabrication, and the connection between the two G electrodes [4; 18]. In any case, the reflection of the RF signals generates a voltage on the electrode two times greater than the set value at the PPG at low frequencies, 1.7 times greater at 12 GHz exhibiting \(S_{11}=-3\) dB, and 1.4-1.5 times greater at 15 GHz exhibiting \(S_{11}=-6\) dB. This enhances the total modulation efficiency while relatively degrading the high-frequency response. For the electro-optic (EO) \(S_{21}\) measurement, a TE-polarized laser light at \(\lambda=1554\) nm was coupled to the SSC through a polarization-maintaining single mode fiber with a lens module. The coupling loss was 3.5 dB, including a 1 dB loss inside the lens module (7 dB in total for the input and output). The total insertion loss from fiber to fiber was approximately 14 dB including a 7 dB on-chip loss. Furthermore, the total loss from the laser source to the output fiber was 21 dB, including losses at the polarization controller and some junctions between fibers. Therefore, the laser power was raised to 16.5 dBm and \(S_{21}\) was measured. The cutoff frequency \(f_{\rm 3 dB}\) was 17 GHz, which is sufficient for the 25 Gbps operation. In general, \(f_{\rm 3 dB}\) is constrained by the \(RC\) time constant. We have reported the calculation of the resistance \(R\) and capacitance \(C\) at the p-n junction using the commercial simulator Lumerical DEVICE in Reference [15]. Modeling the Si layer sandwiched by the SiO\({}_{2}\) BOX layer and top SiO\({}_{2}\) cladding (both insulators), they were calculated to be 96 \(\Omega\) (146 \(\Omega\) including 50 \(\Omega\) internal resistance in the PPG connected in series with the \(R\)) and \(\sim\)60 fF at \(V_{\rm b}=-2.2\times 2\) V, respectively, for the current design. Their product gives \(f_{\rm 3 dB}=18\) GHz, which almost agrees with the measured value. Figure 2: Slow-light characteristics of fabricated device. (**a**) Transmission spectrum; (**b**) \(n_{\rm g}\) spectrum. The \(f_{\rm 3 dB}\) could be constrained by the phase mismatch between the slow light and RF signals. The decreased ratio of the modulation depth, \(\eta(f)\), is expressed as  
\[\eta(f)\ =\ \frac{1}{L}\int_{0}^{L}\cos\ \left\{(n_{\rm g}-n_{\rm RF})\ \frac{2\pi fz}{c}\ +\ \theta_{0}\right\}\ dz \tag{2}\]  
where \(n_{\rm RF}\) is the refractive index for RF signals, \(c\) is the light velocity in a vacuum, and \(\theta_{0}\) is the initial phase mismatch. EO \(S_{21}\) is expressed using electro-electronic (EE) \(S_{21}\) for RF signals as  
\[\mathrm{EO}\ S_{21}\ =\eta(f)\cdot\mathrm{EE}\ S_{21} \tag{3}\]  
For typical traveling-wave modulators, a small phase mismatch is compensated by optimizing the design.