# Fully Reconfigurable Photonic Filter for Flexible Payloads
## 3 Proposed Structure Analysis  
The proposed RF filter architecture is reported in Figure 1. It consists of a single-wavelength continuous-wave (CW) laser, which generates a narrow-linewidth light wave at \(\mathrm{f_{L}}\) (path A in Figure 1). This light wave is then directed to a phase modulator (PM), fed by an RF signal (path B) with frequency \(\mathrm{f_{RF}}\), with two resulting upconverted sidebands (path C) at \(\mathrm{f_{L}}+\mathrm{f_{RF}}\) and \(\mathrm{f_{L}}-\mathrm{f_{RF}}\). One of these sidebands (e.g., at \(\mathrm{f_{L}}-\mathrm{f_{RF}}\)) undergoes filtering via a photonic notch filter, which is the device proposed in the manuscript. At the output of the filter, the optical signal (path D) is O/E transduced by using a photodiode (PD), resulting in the filtered RF signal (path E). The key building block of the RF photonic filter is the optical notch filter. The proposed device comprises two cascaded silicon ring resonators (RRs), both having the same radius R.  
Figure 1: RF photonic filter (PM: Phase Modulator; PD: Photodiode; \(\mathrm{f_{L}}\): laser frequency; \(\mathrm{f_{RF}}\): electrical RF frequency). Insets represent the point-by-point spectra. Each of these resonators can be individually thermally tuned, allowing for reconfigurability in both central frequency and bandwidth. The optical filter is sketched in Figure 2.  
The mathematical model used to characterize the filter response relies on the scattering matrix method [21]. In particular, the scattering matrix of the whole system is calculated by the multiplication of the matrices related to the individual rings. Therefore, the transmission element (\(\mathbb{S}_{21}\equiv\mathrm{T}_{\mathrm{OUT}}\)) results as  
\[\mathrm{T}_{\mathrm{OUT}}\ =\ \left|\mathrm{t}_{1}\cdot\mathrm{t}_{2}\right|^{2} \tag{1}\]  
where \(\mathrm{t}_{1}\) and \(\mathrm{t}_{2}\) are the \(\mathbb{S}_{21}\) elements of the scattering matrix of the RRs. They have been calculated also by using the scattering matrix theory, according to [22]:  
\[\mathrm{t}_{1/2}\ =\ \frac{(1-\mathrm{k}^{2})\cdot\mathrm{e}^{\mathrm{i}2\pi \mathrm{R}\beta}-\sqrt{1-\mathrm{k}^{2}}+\mathrm{k}^{2}\cdot\mathrm{e}^{ \mathrm{i}2\pi\mathrm{R}\beta}}{\sqrt{1-\mathrm{k}^{2}}\cdot\mathrm{e}^{ \mathrm{i}2\pi\mathrm{R}\beta}-1} \tag{2}\]  
where \(\mathrm{k}\) is the amplitude coupling coefficient, \(\tau=\sqrt{1-\mathrm{k}^{2}}\) is the amplitude transmission coefficient, \(\beta\ =\ \frac{2\pi}{\lambda}\mathrm{n}_{\mathrm{eff}}-\mathrm{j}\alpha\) is the propagation constant, \(\alpha\) is the propagation loss, and \(\mathrm{R}\) is the ring radius, assumed as the same for both RRs. The effective refractive index (\(\mathrm{n}_{\mathrm{eff}}\)) has been calculated by using FEM simulations (COMSOL Multiphysics(r) software Release 4.4), also taking into account the Sellmeier relationships and the presence of the TO phase shifters. The contribution of the TO phase shifters to the \(\mathrm{n}_{\mathrm{eff}}\) variation has been estimated through the combination of thermal and optical mode solvers and analyzing both the heat distribution and the efficiency. The presence of TiN TO phase shifters along the whole ring resonators' optical path allows for the reconfigurability of both the central frequency and bandwidth. Further details are reported in Section 3.2. This configuration enables the complete overlap of the two spectral responses \(\mathrm{t}_{1/2}\) when no current is applied to the electrodes, thereby enabling an extended tuning range. The proposed filter is based on a silicon (Si) waveguide, fully embedded in silicon dioxide (SiO\({}_{2}\)), that shows a width (w) of 440 nm and a thickness (t) of 220 nm, to ensure the single-mode operation. The modal analysis of the designed waveguide was made by using FEM simulations (COMSOL Multiphysics(r) software), also taking into account the waveguide dispersion.