*3.3. Graphene Absorption Inside the Fabry Perot Filters: Experimental*

Figure 8a–c shows the comparison between the simulated and measured total absorption curves of graphene inside the three Fabry–Perot filters. The total absorption values (Table 2) include, also, the contribution of SLG absorption (2.3%) and of the FP materials (Figure 5b). A very good agreement between the simulated and measured absorption values was obtained, and only some broadening of the experimental curves was found in all cases. As it can be noticed, the experimental graphene absorption trend is to increase when passing from a symmetric structure to an asymmetric one and it increases further by increasing the reflectance of the bottom mirror. Even though a comparison of the absolute absorption values is not possible due to the different FPs' central wavelengths, the absorption trend strongly suggests that inside a Fabry–Perot cavity a high graphene absorption may be obtained only in asymmetric reflective structures. In our experiments, for example graphene absorption values increased from 39 to 84% when passing from a symmetric to an asymmetric reflective structure. The highest absorption value obtained in the asymmetric reflective FP3 filter is due to the fact that the incident light inside the cavity is reflected a greater number of times which allows to graphene a higher number of multiple absorptions. The number of reflections is in fact related to the Fabry–Perot finesse which is defined by [68,88–90]:

$$F\_{\rm s} = \frac{\pi\sqrt{R}}{1 - R} \text{ and } F\_{\rm as} = \frac{\pi\sqrt[4]{R\_{\rm Top} R\_{\rm Bottom}}}{1 - \sqrt{R\_{\rm Top} R\_{\rm Bottom}}}$$

Above, the left-hand side formula represents the finesse of the symmetric FP filter (FP1), where *R* is the reflectance of the top and bottom mirrors, while on the right-hand is the finesse of the asymmetric filters (FP2 and FP3). In our FP filters, the finesse increased from 21 to 31.5 when passing from the symmetric structure of Figure 1a (FP1) to the asymmetric reflective structure of Figure 1c (FP3).

Absorption of unpatterned and undoped single layer graphene up to 84% has been never reported experimentally in a Fabry–Perot filter. Comparable or even higher absorption values have instead been obtained by other authors by using graphene coupled to non-planar resonant plasmonic structures [45,91,92].

Enhancing the light-matter interaction in single-layer graphene by the use of optical microcavity requires no stringent constrains on the graphene electrical properties, such as the sheet resistance, and may find application in a variety of graphene-based devices, such as optical absorption modulators, light emitters, and optical attenuators, and provides new routes to graphene photonics for applications in spectroscopy, communications, sensing, and security.

Results, validated here at NIR and MIR wavelengths, may be applied in principle at different wavelengths ranging from Vis to THz by properly choosing the materials of the layers and of the substrate. Moreover, as obtained by our previous simulations, modification of the filter amplitude and bandwidth may be achieved by: (i) using multilayer graphene instead of single layer graphene [68], (ii) using two or more SLGs positioned in the filter where the electric field has its maxima [69], and (iii) increasing the number of cavities of the multilayer structure [68].

**Figure 8.** Comparison between simulated and experimental absorption curves of single layer graphene inside the Fabry–Perot filters: (**a**) symmetric FP1, (**b**) asymmetric FP2, and (**c**) asymmetric reflective FP3.

Figure 9a shows, as an example, the absorption of a single layer graphene, of a doublelayers graphene (DLG) and of a five-layers graphene (MLG) embedded in the symmetric FP1. As it can be noticed, absorption bandwidth increases by increasing the number of layers, while absorption amplitude increases in DLG and decreases in MLG. The latter behavior can be attributed to the higher intrinsic absorption of MLG which reduces the reflections of the light inside the optical cavity, and to the fact that MLG perturbs largely the optical quality of the Fabry–Perot cavity [68].

**Figure 9.** Absorption of graphene layers inside the FP1 as a function of number of graphene layers (**a**) and absorption of a single layer graphene in a single Fabry–Perot cavity and in a dual Fabry–Perot cavity (**b**).

Figure 9b shows the effect of the number of Fabry–Perot cavities on the absorption of a single layer graphene [68], showing a higher graphene absorption inside a dual cavity Fabry–Perot filter.

Results are in accordance with what has been reported by other authors. In [46], Xiao et al. have numerically modeled, in a structure different from ours, the bandwidth absorption as a function of the number of graphene layers finding a broader absorption band by increasing the number of graphene layers from 1 to 7. In [50], Ferreira et al. have numerically simulated that absorption of single layer graphene increases by using a dual cavity Fabry–Perot instead of a single one.

Graphene absorption discussed so far was obtained in the case of a TE polarization. For the optimized FP structure, i.e., the FP3 filter, the simulation was carried out, also, in case of TM polarization. Figure 10 shows the comparison between the simulated absorption of SLG for TE and TM polarizations as a function of the incident angle. As it can be noticed, in case of TE polarization the absorption maximum is reached for incident radiation, while for TM polarization it occurs at a higher incident angle of 60◦ .

**Figure 10.** SLG optical absorption as a function of wavelength for different incident angles at (**a**) TE and (**b**) TM polarization.

#### **4. Conclusions**

The increase of single layer graphene absorption obtained by exploiting the electric field enhancement inside a resonant optical cavity was modeled and experimentally demonstrated in three different Fabry–Perot filters with central wavelengths varying in the NIR-MIR spectral ranges. The Fabry–Perot filters were fabricated by radiofrequency sputtering, and consisted of alternate quarter wave thick Si and SiO<sup>2</sup> layers. Results demonstrated that graphene absorption greatly increases when graphene is embedded inside an asymmetric Fabry–Perot structure, reaching its maximum in case of a reflective Fabry–Perot filter. SLG properties were preserved during the sputtering process by applying a thin, slowly evaporated MgF<sup>2</sup> layer, allowing the effective embedding of graphene inside thick monolithic structures fabricated by conventional PVD techniques. Such a high graphene absorption discloses exciting potentiality for exploitation of 2D materials in new optoelectronic devices for application in the NIR-MIR spectral range.

**Author Contributions:** Conceptualization, M.L.G. and A.N.; supervision of the study, M.L.G.; simulation, A.N.; investigation, A.N., M.L.G., N.L. and L.L.; data curation, M.L.G. and A.N.; writing—original draft preparation, M.L.G.; writing—review and editing, M.L.G., A.N. and N.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Data are available on request from the corresponding author.

**Acknowledgments:** Authors acknowledges the support from the International Centre for Theoretical Physics, ICTP TRIL fellowship number 3356. Authors thank Angelo Gentili for his excellent technical maintenance of the sputtering and evaporation systems.

**Conflicts of Interest:** The authors declare no conflict of interest.
