## 2 Materials and Methods  
A Plasmalab 100 ICP 180 from Oxford Instruments Plasma Technologies (Bristol, UK) at a substrate temperature of 80 \({}^{\circ}\)C and an inductively coupled plasma (ICP) generator at a radio frequency (RF) of 13.56 MHz with an optional additional RF field between the electrodes is the modular plasma processing system employed in this study. Two-inch boron-doped (100) silicon wafers with a thickness of 280 \(\upmu\)m are used as substrate. We implemented different parameters of gas flow ratio, RF power, and chamber pressure, for growth of thin films of silicon-rich SiN\({}_{\mathrm{x}}\) at a deposition duration of 15 min. The precursor gases used in this deposition are N\({}_{2}\) and SiH\({}_{4}\), avoiding the absorption centres that NH\({}_{3}\) creates [3]. For every recipe, refractive index (\(n\)), extinction coefficient (\(k\)), and deposition rate (DR) are determined using ellipsometry (Sentech 850) at an incidence angle of 70\({}^{\circ}\). In this methodology, a Cauchy model is employed in order to determine these parameters [21]. Then, the roughness of the film is measured using a Dektak 6M Stylus Profiler, as well as the curvature of the substrate before and after deposition, that allows the evaluation of the stress gradient of the layer using the bending plate method [22]. Once the parameters of refractive index, roughness, and stress are determined, applying different processes of etching and evaporation techniques, following a similar protocol to the one described in [12], we fabricated on wafer hundreds of MEMS-FP cavities with 9 different architectures. Each of these filters consists of a plane concave cavity formed by a bottom and a top DBR with two different electrodes on top of each mirror. A 6-layer pair of quarter-wavelength silicon-rich SiN\({}_{\rm x}\) and SiO\({}_{\rm y}\) DBR deposited over a double side polished substrate of boron-doped (100) silicon wafer with a thickness of 280 \(\upmu\)m forms the plane part of the cavity. Over this bottom reflector is deposited a second DBR, in this case with 6.5-layer pairs and a concave shape. The curved structure forms as a result of the compressive stress of the layers and a chromium sacrificial layer (of height \(h_{0}\), c.f. Figure 1a) that separates both reflectors during the fabrication process resulting in the final air gap (\(L_{cav}\)). The SiO\({}_{\rm y}\) layers of both DBR were deposited with a flow of precursor gases 12:3 sccm SiH\({}_{4}\):N\({}_{2}\)O, an IPC power of 500 W, an RF power of 75 W at 7.5 mTorr of chamber pressure. SiO\({}_{\rm y}\) features a DR of 15.5 nm/min, a refractive index at 1550 nm of \(n=1.4476\) and a compressive stress of 130 MPa. The reflection band of the DBR is measured using ellipsometry at 40\({}^{\circ}\), the minimum angle achievable by the ellipsometer in order to have a measurement as close to 0\({}^{\circ}\) as possible. Figure 1a shows a schematic cross-sectional image of a MEMS-FP cavity. The geometry of the FP-cavity bottom and top electrode are illustrated in Figure 2 where the width (\(W\)) and length (\(L\)) of the arms are altered for every architecture while the central ring in the top electrode that connects the four arms keeps a constant external radius of 40 and an internal radius of 25 \(\pm\) 1 \(\upmu\)m. This hole is the optical window. Using the Dektak stylus profilometer, we measured the curvature of every architecture. Then, the nine architectures are simulated in Comsol Multiphysics in order to derive from the measured curvature the required the values in [PERSON]'s modulus, \(E\), of the silicon-rich SiN\({}_{\rm x}\) deposited in the DBR assuming a Poisson's ratio of 0.27 [11]. A CTL 1550 tunable diode laser from Toptica Photonics (Grafelfing, Germany) with a tuning range from 1510 to 1630 nm was used as laser source for the tuning characterisation of one of the designs.