*2.1. Fabry–Perot Filter: Simulation*

A Fabry–Perot optical filter consists of a cavity separated by two flat and parallel high reflecting mirrors where light experiences multiple reflections. Simulation of FP filters was carried out by using the Wave Optics Module of Comsol Multiphysics software. Our Fabry–Perot filters consisted of a sequence of alternate quarter wave thick high refractive index Si (H) and low refractive index SiO<sup>2</sup> (L) layers, for use in the Near to Mid IR. Three different Fabry–Perot filters were simulated: (i) a symmetric FP filter with identical top and bottom mirrors (FP1), (ii) an asymmetric FP filter where the bottom mirror had a higher reflectance due to a larger number of layers with respect to the top mirror (FP2), and (iii) an asymmetric reflective FP filter obtained by further increasing the bottom mirror layers (FP3). FP1 and FP2 filters worked in transmissive mode, while FP3 worked in reflective mode, due to the high reflectance of the bottom mirror (99.7%). The structure of FP1, FP2, and FP3 filters was the following: (i) air/HLH LL HLH/sub, (ii) air/HLHL HH LHLHLH/sub, and (iii) air/HLH LL HLHLHLHLH/sub, respectively. For FP1 and FP3 the cavity was (LL), i.e., constituted by a half wavelength thick SiO<sup>2</sup> layer, for FP2 the cavity (HH) was made by a half wavelength thick Si layer. This last FP structure, a bit different from the other two, was chosen because of its particular distribution of the electric field characterized by two maxima. A single layer graphene (SLG) was embedded inside the FP structure and located at the position where the electric field was maximum to enhance its absorption. SLG was covered by a 30 nm MgF<sup>2</sup> layer, meant to protect graphene during the later sputtering deposition. Figure 1 shows the three FP filters embedding SLG, and Figure 2 shows the

simulated reflectance of the bottom mirrors in the three cases, R1, R2, and R3, R1 also has the reflectance of the top mirror, being identical for the three filters.

**Figure 2.** Simulated reflectance curves of the bottom mirrors shown in Figure 1. R1, R2, and R3 are the reflectance of bottom mirrors of FP1, FP2, and FP3, respectively.

λ λ Figure 3a shows the simulated transmittance (T), reflectance (R), and absorption (A) of the FP1, FP2, and FP3 filters, centered at λ = 2315, 4342, and 3150 nm, respectively. Figure 3b shows the electric field distribution inside the filters. The simulation has been carried in case of a TE polarized light at normal incident angle.

Table 2 reports the simulated and experimental absorption values of SLG embedded in the three FP filters and the value of electric field (EG) where SLG is positioned. The central wavelength of the Fabry–Perot filters is different for the three cases because it results from different experiments, however, central wavelength position in the considered wavelength range, only causes small variations of T, R, and A due to the wavelength dependence of Si and SiO<sup>2</sup> optical constants. As we can observe from Table 2, the highest graphene absorption is obtained in the structure where the electric field is maximum.

**Figure 3.** (**a**) Simulated transmittance (T), reflectance (R), and absorption (A) of Fabry–Perot filters FP1, FP2, and FP3; (**b**) electric field distribution inside the Fabry–Perot filters. SLG position is indicated in the three filters by a black line positioned at electric field maximum (red color).


**Table 2.** Graphene-based Fabry–Perot characteristics. Simulated and experimental absorption and electric field value at SLG position (maximum of electric field).
