*3.2. Graphene Quality Assessment*

The quality of graphene was checked after each experimental step, i.e., after MgF<sup>2</sup> evaporation, after SiO<sup>2</sup> sputtering on MgF2/graphene layers and after the whole filter fabrication process.

Initially, for the growth of the first 30 nm of SiO<sup>2</sup> layer directly onto SLG, a mild sputtering condition was applied by increasing the deposition pressure and decreasing the radiofrequency power. Notwithstanding, graphene damage could not be avoided, as demonstrated by Raman analysis (not shown here), and a further graphene protection was necessary. Therefore, to avoid the sputtering of the SiO<sup>2</sup> layer directly onto graphene, a thin protective MgF<sup>2</sup> layer was evaporated prior to the sputtering of SiO2. We have chosen MgF<sup>2</sup> due to its refractive index and extinction coefficient values, very close to those of SiO2. This is necessary since MgF<sup>2</sup> becomes part of L (the low index layer) of the multilayer stack. Other fluoride materials could have served the purpose. Furthermore, the evaporation of the MgF<sup>2</sup> layer required some care since we observed that if evaporation occurred at a high rate, graphene defects band D increased. Figure 7a compares the Raman spectra of pristine graphene and graphene after evaporation of MgF<sup>2</sup> at high (2 Å/s) and low (0.1 Å/s) evaporation rates. Only the latter case left graphene unchanged. The effect of deposition rate of SiO<sup>2</sup> was preliminarily evaluated in experiments with ML graphene with unoptimized MgF<sup>2</sup> deposition (curve of Figure 7b). The effect of the optimized, low power and high pressure SiO<sup>2</sup> deposition on SL graphene can be seen in the red and blue curves of Figure 7c. Here the comparison of the red (Raman from the back of the peeled stack) and blue curves (Raman from the top through the SiO2), suggests that some extent of the modifications of the D and 2D bands by SiO<sup>2</sup> is not due to lattice damage but to optical and electronic "proximity" influences on graphene by the stack.

Despite the increase of the defectiveness D band, the optical properties of graphene were preserved, as shown by the comparison of UV-Vis absorbance curves of pristine graphene transferred on a quartz substrate and graphene after MgF<sup>2</sup> and SiO<sup>2</sup> deposition (Figure 7d). Moreover, two-point probe measurements of SLG resistance prior and after the MgF<sup>2</sup> and SiO<sup>2</sup> sputtering confirmed that graphene resistance was almost unaffected by these processes. SLG sheet resistance increased, in fact, of less than 10% with respect to the initial value of about 800 Ω/sq.

Ω

**Figure 7.** Raman spectra of: (**a**) SLG showing the effect of MgF<sup>2</sup> evaporation rate, (**b**) MLG showing the effect of SiO<sup>2</sup> sputtering conditions, and (**c**) SLG showing the effect of insertion inside the Fabry– Perot filter. (**d**) Comparison of absorbance of SLG pristine (as transferred on quartz substrate) and after MgF<sup>2</sup> evaporation and SiO<sup>2</sup> sputtering.

1

The results obtained after several attempts to not damage graphene can be summarized as following: (i) a thin MgF<sup>2</sup> layer was essential to protect graphene, (ii) evaporation of the thin MgF<sup>2</sup> layer should occur with a very low rate, (iii) the thin MgF<sup>2</sup> layer alone was not enough to avoid damage if sputtering process is too energetic, (iv) combination of evaporated MgF<sup>2</sup> and mild sputtering condition during SiO<sup>2</sup> growth produces only an increase in the graphene defective band (D band) but preserves the optical and electrical properties of pristine graphene, and (v) we could not quantify the damage of graphene inside the stack due to possible proximity effects by the embedding layers on D and 2D bands.
