## 2 Background  
### Silicon Photonics Modulators  
In SiPh technologies, an optical modulator is usually realized by embedding a pn-junction inside a waveguide [18]. It is well known that the presence of free charge carriers, i.e., electrons and holes, can change the refractive index of a semiconductor material [19]. By varying the pn-junction bias voltage, free-carriers' distributions change across the waveguide, and the phase velocity of the propagating optical mode is modified. Depletion-mode operation ensures high-speed phase modulation since only majority-carriers' dynamics are involved [20]. Amplitude modulation can be achieved by inserting phase shifters into interferometric, e.g., MZMs [21], or resonant architectures, e.g., RMs [22; 23]. Information is indeed typically encoded in the light intensity, and the simple on-off keying (OOK) is one of the most adopted modulation formats, although more complex schemes are emerging, e.g., PAM-4 [24]. In order to make efficient designs for the electronic driver, it is important to carry out an electrical equivalent circuit of the phase shifter under study. The reverse-biased diode can be represented by a voltage-dependent junction capacitance in series with the access resistances associated to the ohmic conduction through the anode and cathode doped slabs. The junction capacitance per phase shifter length \(C_{l}\) is typically around hundreds of \(\mathrm{fF/mm}\), while the per-unit-length resistance \(R_{l}\) is normally about a few \(\Omega\cdot\mathrm{mm}\). For modulation wavelengths \(\lambda_{\mathrm{rf}}\), which are greatly larger than the device dimensions, i.e., \(l\ll\lambda_{\mathrm{rf}}\), the appropriate circuit model for a phase shifter of length \(l\) is depicted in Figure 0(a). In the limits of this approximation, the overall device behavior is captured by means of lumped-elements (LEs), and the intrinsic (\(-3\,\mathrm{dB}\)) bandwidth results [20]:  
\[f_{\mathrm{3 dB}}=\frac{1}{2\pi\left(R_{l}C_{l}+Z_{\mathrm{drv}}lC_{l}\right)} \tag{1}\]  
Assuming a purely real driver output impedance \(Z_{\mathrm{drv}}\) and neglecting all the parasitic contributions. Being proportional to the amount of charge carriers that are swept in and out the active device region, \(C_{l}\) is directly related to the phase shifting efficiency, which is commonly quantified by the \(V_{\pi}l\) figure of merit. Since the free-carrier effects are weak in silicon, \(V_{\pi}l\) typically ranges between \(10\,\mathrm{V}\cdot\mathrm{mm}\) and even \(40\,\mathrm{V}\cdot\mathrm{mm}\) depending on the implant conditions [26]. Because the available voltage span is typically no more than a few volts in deep sub-\(\upmu\mathrm{m}\) CMOS technologies, phase shifters are normally operated with voltage swings substantially lower than the rated \(V_{\pi}\)[27]. Interference-based modulators are then made a few millimeters long to accumulate enough phase shift and provide reasonable optical extinction ratios (ERs). As a consequence, the total capacitance \(lC_{l}\) is enlarged, and \(f_{\mathrm{3 dB}}\) is diminished according to Equation (1). On the other hand, compact devices, such as RMs, present small lumped capacitances and can guarantee high cutoff frequencies from a purely electrical point of view, i.e., ignoring photon lifetime-related effects. To cope with the capacitance-limited response of long modulators, traveling-wave (TW) devices are usually conceived. By designing phase shifters as transmission lines (TLs), the capacitive loading could be accurately \"mixed up\" with the inductive contribution of the coplanar electrodes, and the whole device, when properly terminated, globally appears as a (resistive) load equal to the TL's characteristic impedance \(Z_{0}\) (Figure 0(b)). By choosing \(R_{L}\approx Z_{0}\approx Z_{\mathrm{drv}}\), impedance matching can be theoretically achieved (neglecting parasitics), and by making them equal to \(50\,\Omega\), the compatibility with common testing instrumentation can be maintained.