Main principles of passive devices based on graphene and carbon films in microwave—THz frequency range

Abstract. The ability of thin conductive films, including graphene, pyrolytic carbon (PyC), graphitic PyC (GrPyC), graphene with graphitic islands (GrI), glassy carbon (GC), and sandwich structures made of all these materials separated by polymer slabs to absorb electromagnetic radiation in microwave-THz frequency range, is discussed. The main physical principles making a basis for high absorption ability of these heterostructures are explained both in the language of electromagnetic theory and using representation of equivalent electrical circuits. The idea of using carbonaceous thin films as the main working elements of passive radiofrequency (RF) devices, such as shields, filters, polarizers, collimators, is proposed theoretically and proved experimentally. The important advantage of PyC, GrI, GrPyC, and GC is that, in contrast to graphene, they either can be easily deposited onto a dielectric substrate or are strong enough to allow their transfer from the catalytic substrate without a shuttle polymer layer. This opens a new avenue toward the development of a scalable protocol for cost-efficient production of ultralight electromagnetic shields that can be transferred to commercial applications. A robust design via finite-element method and design of experiment for RF devices based on carbon/graphene films and sandwiches is also discussed in the context of virtual prototyping.


Introduction
Electromagnetic (EM) properties of graphene and its ability to absorb light and EM radiation of longer wavelengths is very dependent on the frequency range of EM wave, more precisely on what is the dominant contribution to graphene ac conductivity, either inter-or intra-band transitions [1,2]. It is already well-known that one monolayer of graphite is absorptive in optics, providing  = 2.3% absorption of light of visible spectra, which comes from its conductivity originated by inter-band transitions. When one switches to the lower frequencies, THz-subTHz ranges, the inter-band transitions could lead to a much higher absorbance, at the level of 15-20% per plane, depending on the graphene conductivity caused in its turn by the doping level [3,4].
Graphene, being light, flexible, shatter-proof is very attractive for conventional electronic nanodevices as well as for being used in electronic components in post-silicon age. Its ability to absorb large amount of RF radiation might be used for the design of different passive devices.
The purpose of the present paper is to demonstrate some of these possibilities while underlying them with the main physical principles of such devices operation on the basis of graphene.
One more important goal is to introduce other conductive carbonaceous films that could be produced or transferred more easily than CVD graphene. These films could swap graphene in the same applications as building blocks of passive RF devices.

Main principles
In the following, the words "thin conducting layer" will refer to carbon-based thin film. It can be pyrolytic carbon a few nm thick. Ultimately, it can also be an individual graphene plane or several graphene planes. In the later case, the conductance scales linearly with the number N of graphene planes when no crystallographic correlation is assumed between the successive atomic lattices. This property is realized in a random stacking of CVD graphene planes [5] and in graphene/PMMA multilayers [3]. In these multilayers, the PMMA spacers used for the transfer of graphene are 0.5 µm thick or less. They are so thin at the wavelength scale that play no role in the electromagnetic properties of the system and can safely be forgotten for frequencies up to the THz range.
When a thin -at the scale of the skin depth -conducting layer is located at the interface between two dielectric media, the component of the electric field, which is parallel to the interface induces a current density (A/m) J = σE∥, where σ is the conductance (sheet conductivity) of the layer. The current density, in its turn, is responsible for a discontinuity of the parallel component of the magnetic field according to the equation n×(H + ∥ -H -∥) = σE∥ where n is the local normal unit vector. By contrast, the normal component of the electric field is continuous across the interface.
From there on, the Fresnel formula for the reflectance R, the transmittance T, and the absorptance A of a plane wave incident on the interface can readily be derived for the geometry illustrated in where = − can be an imaginary number when the incidence angle θI exceeds the critical angle. The formulas for the transverse magnetic (p) polarization follow from the above by the substitution of 1/cos for all the cosine functions. When the sample is placed inside a waveguide and illuminated by a pure guided wave, all the cosine functions must be set to unity (incidence at angle with = 1 − ( ⁄ ) )) and the indices of refraction must be reduced by the factor of 1 − ( ⁄ ) , where ωc is the cut-off frequency [4]. In free space and normal incidence, the absorptance reaches a maximum value nI/(nI+nT), when σ is real and σ/ϵ0c = nI+nT [7].
If the thin conducting layer is composed of N non-interacting graphene planes, the conductance is proportional to N. Accordingly, an optimum value of N exists for which the absorptance is maximum. For GHz frequencies, this property is best demonstrated in a waveguide. The condition for maximum absorption writes ⁄ = 2 1 − ( ⁄ ) for an air filled waveguide.
The optimum number of graphene planes is therefore smaller than it would be in free space (ωc →0) due to the square root factor. In the experiment of Ref. [3], this factor represents 0.72 at 30 GHz for the TE10 mode of the rectangular waveguide used.
In free space, assuming σ real, the small-angle variation of the absorptance is given by for s (+ sign) and p (-sign) polarization. As and Ap behave diametrically opposite with increasing incidence angle θI, raising or decreasing depending on whether the conductance σ is smaller or larger than (nI + nT)(2nI/nT -1)ϵ0c. For small incidence angles, the average absorptance (As+ Ap)/2 (non polarized radiation) is constant up to fourth order in θI.   Fig. 2c). The p transmittance increases monotonously vs θI (dashed line curve in Fig. 2d). The reflectance and transmittance for the s polarization behave diametrically.
As schematized in Fig. 2b, a thin conducting layer of the kind discussed here becomes a polarizer at grazing incidence [8]: almost 100% of the reflected radiation is s polarized and a big fraction of the transmitted wave is p polarized. illustrates how such a conducting layer (thick green line) may behave as a polarizer at grazing incidence. The blue full lines and red dashed lines correspond to the s and p polarizations, respectively. The black dot-dashed lines correspond to non-polarized radiations.

Equivalent circuit approach
For most applications, graphene has to be supported by a substrate, which thickness can not be neglected as compared with the wavelength of interest. The system graphene plus substrate can be oriented with the thin conducting layer facing either the incidence medium or the emergence medium. One may, therefore, consider exploring the properties of a heterostructure composed of a thin conducting layer located between two dielectric slabs as illustrated in Fig. 1(b).
From the electromagnetic point of view, this system is equivalent to an electrical circuit [9]. The equivalent circuit for the s polarization is shown in Fig. 3(a). The generator accounts for the power that is required to maintain the electromagnetic radiations in a stationary state in spite of the dissipation by Joule effect in the thin conducting layer. For reasons detailed in the Introduction, it is desirable to maximize the absorption over the reflection in electromagnetic shielding layers, very much like to radar absorbing media. One needs therefore to design the heterostructure of Fig. 1 where k0 = ω/c. The non-standard formalism used for the elements of the circuit of Fig. 3(a) has the advantage to yield an easy generalization of the expression of Y1 and Y2 in the form of a continued fraction when the system has an arbitrary number of dielectric layers [10]. In normal incidence, (eq. 3) can be used irrespective of the polarization. The case where there is just single dielectric layer, oriented upwards or downwards with respect to the incident radiations, is now analyzed under the explicit assumption that both ϵ1 and ϵ2 are real.

2.2.a Graphene on a substrate facing the incidence medium.
This case corresponds to d2 = 0 in Fig. 1(b). At normal incidence, the expression (eq. 3) of YN simplifies in For real σ, the admittance matching condition requires YN to be real, which is realized when either sin ϕ1 = 0 (half-wavelength blade) or cos ϕ = 0 (quarter-wavelength blade). The first case is equivalent to the geometry of Fig. 1(a) with its maximum absorptance condition σ/ε0c = nI + nT. The second case demands σ/ε0c = ϵ1/nI + nT for optimum absorption and leads to an absorptance larger than in the first case, provided ϵ1>nI². The optimum value of A obtained after additional calculations is Aopt = ϵ1/(ϵ1+nInT). An illustration of this effect is described in Section 4.

2.2.b Graphene on a substrate facing the emergence medium.
This case corresponds to d1 = 0 in Fig. 1(b). The expression of YN is the same as in (eq. 4) after the substitution of the index 2 for the index 1 and nT for nI. Still for a real σ, the condition sin ϕ2 = 0 is equivalent to the geometry of Fig. 1(a). The condition cos ϕ2 = 0 is more interesting and leads to the condition of maximum absorption σ/ε0c = nI + ϵ2/nT, with an absorptance larger than the first case provided ϵ2<nT². Here, the optimum value of the absorptance is Aopt = nInT/(ϵ2+nInT).
It can approach 100% when nT→∞. The Salisburry screen invented in 1952 pertains to this geometry [11]. In this device, the emergence medium is a metal (nT→∞) thicker than its skin depth, separated from the thin conducting layer by a non-absorbing dielectric slab of thickness d2. The absorptance is maximum for frequencies such that the effective wavelength in the dielectric slab is a multiple of 4d2 (cos ϕ2 = 0, or more precisely Im YN = -Im σ when σ is complex).

Synthesis of carbonaceous materials
Graphene, PyC, GrPyC and GrI are all fabricated by chemical vapor deposition at ~1000 °C temperature regime using methane precursor. Graphene, GrPyC and GrI were grown on a copper foil whereas PyC was grown directly on a silica substrate. Typically, a large area monolayer graphene is grown on a copper substrate [12]. However, copper can be used as a substrate for thicker graphitic film synthesis as well. Annealing the copper substrate or increasing the amount of methane increase the carbon film thickness. This will lead to synthesis of GrPyC and GrI films. Without catalyst metal (i.e. the process on a silica substrate), an amorphous carbon film is produced with dominating sp2 hybridization [13]. This will lead to ultra-thin and uniform PyC film formation [14].
The experimental receipts for graphene, PyC, GrPyC and GrI are similar to each other. All materials demonstrated here are fabricated in conventional hot wall CVD by using methane precursor. Pyrolytic carbon was grown on a silica substrate at dynamic temperature range starting at 700 C. First the CVD chamber was heated to 700 C in hydrogen atmosphere. Then the temperature was increased in CH4 atmosphere (static 25 mBar) to 1100 C and after 5 minutes the CVD chamber was cooled down to 700 C and the methane atmosphere was replaced with hydrogen (for more details see Ref. [15]). Graphene was grown on a copper foil (99,8 % pure) at 950 C temperature. First the copper foil was heated and annealed (20 min) in H2 atmosphere (5 sccm, 0,5mBar). The graphene film was grown in 950 C using 5sccm H2 + 5 sccm CH4 flow for 20 minutes. GrPyC was grown on copper foil (99,8 % pure) in 1000 C temperature in static CH4 atmosphere (25 mBar) in 30 minutes (see Ref. [16] for details). This process produced graphitic carbon film about 8 nm thick. GrI sample was also grown on a copper foil (99,8 % pure). The copper substrate was first annealed in 1000 C temperature in H2 atmosphere (5 sccm, 0.5 mBar) for one hour. After this the graphitization was done in static atmosphere using methane and hydrogen gas mix (with 1:1 ratio in 8 mBar pressure) for 20 minutes. All the samples were cooled down to room temperature in static H2 atmosphere, ~5 mBar (overnight).
The growth of PyC on a dielectric substrate does not require a metallic catalyst. In about 1000 C temperature the spontaneous methane decomposition leads to C2 and eventually aromatic C6 hydrocarbon species formation. These molecules land on a dielectric surface and evolve forming nanosize graphitic flakes that intervene and form a continuous film. Although the PyC film is continuous and dominated by sp2 hybridized carbon, the film is very amorphous. Therefore, many properties like electrical conductivity and nonlinear optical effects are suppressed in comparison to graphitic carbon films [15,17].
Low carbon solubility in copper is responsible for graphitic monolayer growth on the surface of a copper substrate in CVD process. This technique is widely explored and is recognized as one of the most promising method for large area graphene synthesis. The process is based on surface catalysis. The methane molecule is decomposed at the surface of a copper substrate and, due the low carbon solubility of copper; the carbon atom stays at the surface of the copper substrate.
Eventually, the surface of copper is covered by sp2 hybridized monolayer of carbons. Once the copper surface is coated with graphene, the catalytic growth of graphite is almost prevented.
However, increasing the amount of methane during the process will cause template graphitic film growth on top of the monolayer graphene resulting a thin graphitic film. Moreover, if the copper substrate is annealed for one hour, we observed graphitic islands grown on a monolayer graphene. These graphitic islands appeared as either round areas or stripes (see Fig. 4). The process temperature of 1000 C is below melting point of copper (1084 C) but is 11 enough for surface melting. Prolonged annealing time may increase the depth of melted copper slightly increasing carbon solubility in copper resulting multilayered graphitic areas.
A GC was fabricated by spin coating carbon based photoresist (nLOF-AZ2070 diluted with AZ EBR solvent with 1:4 ratio) on a substrate. After the substrate is coated by a resist layer, it is baked on a hot plate (110 °C/ 1 min) and then pyrolyzed in a CVD system. The thickness ratio of the original photoresist film and the GC is 1:10, i.e. the thickness of a 300 nm thick photoresist film led to a 30 (± 3) nm thick GC after the pyrolysis. The thickness of the GC film can be vary by changing the thickness of the photoresist film [18,19].

Carbon film transfer and characterization
Graphene, GrPyC and GrI grow on a Cu substrate and need to be transferred for further use.
GrPyC is mechanically much more robust than graphene, and therefore survives the transfer process from a Cu substrate without a template polymer layer [16] typically used in graphene transfer process to protect and hold graphene.
Transfer process of graphene and GrI is done by using Poly(methyl methacrylate) (PMMA) as a support. First the the PMMA layer is spin coated on a graphitic film/copper substrate and baked in about 60 C for 10 minutes. After baking, the backside carbon of the sample is removed by oxygen plasma etching (20 sccm/100 W/2 min) and the copper substrate foil is then wet etched in ferric chloride. The PMMA/graphitic film is next rinsed in water two times to remove FeCl3 remains and deposited then on a dielectric substrate. Remaining water is evaporated by baking the sample on a hot plate in mild temperature.
When all water is dried from the sample the PMMA support is removed by acetone bath (overnight) and then rinsed in isopropanol and water. This procedure will remove most of the PMMA but sometimes leave remains that can be seen in Fig 4. The PyC film is deposited directly on a dielectric substrate and is usually not needed to be transferred. However, should the film be lifted out from the substrate, it is noteworthy that the adhesion between the PyC film and its silica substrate is rather weak. Depositing PMMA on a substrate and then gently placing the sample in water sometimes lifts the PyC/PMMA film from the substrate. From the water surface, the PyC/PMMA film can be deposited on an arbitrary substrate and the PMMA can be removed similarly as for graphene transfer.
After the carbon films are transferred on the desired substrate, it is important to characterize them. In our experiments, we used Raman spectroscopy, which is well-known technique to identify the carbon allotropes. Moreover, optical transmission spectroscopy was used to provide some understanding of the linear optical properties of the fabricated films. A Raman spectrum of a graphitic carbon material is governed by three main peaks. Those are D (located ~1350 cm-1), G (located ~1582 cm-1) and 2D (located ~2700 cm-1) peaks. Generally, the D peak is a breathing mode, activated by a disordering in the graphite lattice while the longitudinal mode G peak is a trace of a graphitic sp2 hybridization of carbon atoms. The 2D peak is a second harmonic of D peak but does not require disorder.
Raman spectra of the carbon films were measured by a Via Raman microscope with 514 nm excitation wavelength. Fig. 5 shows Raman spectra of graphene, GrPyC, GrI and PyC. As seen, graphene and GrI are almost comparable with their small D peak but strong and rather narrow G and 2D peaks. The small D peak is expected to originate from the bi-/multi-layer islands in graphene and GrI. In GrPyC, the D peak is significantly stronger in comparison to graphene and GrI because the multilayer areas are grown by template manner without catalyst and this increases the amount of disorder in the material. From the Fig. 5 (a) it can be seen that the PyC film is already a very amorphous material with wide D and G peaks and has negligible 2D peak.
Therefore, it can be summarized that graphene, GrPyC and GrI are rather crystalline materials, while PyC is amorphous graphitic carbon. A careful analysis reveals that there are practically no differences in the Raman spectrum of the PyC and GC [19].
The transmittance spectra of the carbon films were measured by a Perkin Elmer lambda-18 spectrometer in a wavelength range from 250 nm to 750 nm. They are shown in Fig. 5. As it can be seen, all of the films have almost constant transmittance from 500 nm to 800 nm and a transmittance dips at 260 nm. As it is shown in Raman spectra of graphene, GrPyC and GrI possess higher crystallinity in comparison to PyC. The absorption of the GC is about 10 % higher in comparison with PyC of the same thickness [19]. Therefore, the drop in the transmittance for GC and PyC is wider in comparison to other films.

Sandwich structures fabrication
The sandwich structure of the multilayered graphene/graphitic film, described theoretically in chapter 2 is fabricated by using the PMMA layer as a dielectric spacer between the carbon films.
The sample fabrication process follows the transfer process to the point where PMMA/graphene film is deposited on a substrate. When PMMA/graphene sample is on a substrate, instead of removing PMMA in acetone, another PMMA/graphene film is deposited on top of the first layer.
By this technique multilayered sandwich structures can be easily stacked with a desired number of layers (see Fig. 6).

Electromagnetic measurements
The EM interference (EMI) shielding ability as ratio of transmitted/input, S21, and reflectance as ratio of reflected/input, S11, signals were measured at 26-37 GHz by a scalar network analyzer R2-   By selecting the optimum substrate thicknesses and number of layers (therefore the overall conductance), it becomes possible to rise the absorption in the microwave range at 80-95% levels. In this case, as follows from the theoretical description, one of the central parameters defining the electromagnetic properties of the sandwich system is the conductance of the original carbon film, which in its turn is influenced by many factors.
The dependence of the scattering matrix elements of structures consisting of one carbonaceous layers is shown in Fig. 8. Analyzing the results collected in Figure 8, one may conclude that graphene is the best candidate for EMI shielding in case optical transparency is also needed for instance for electromagnetic interference (EMI) shielding of windows of optoelectronic devices. Moreover, the contribution of reflection (S11) in graphene/PMMA sandwich consisting of one graphene monolayer in microwave range is the smallest in comparison with all other carbonaceous films (46% vs 52% for GrI, which is closest to graphene regards to optical density).
At the same time, while not as transparent as graphene is, thin PyC films (5 nm thick), GrI and GrPyC (8-12 nm thick) could be interesting alternative to graphene, as they are still transparent enough to protect optoelectronics devices and provide better (18-20% vs 15% for one layer structure) shielding ability in microwaves.
Thicker PyC and GC, being 25-30 nm thick, absorb already large amount of visible light, close to 50%, and might be used for other tasks where optical transparency is not a must (such as EMC tasks for electronic devices working in THz and sub-THz frequencies).
Although PyC and GC being 25-30 nm thick are not transparent in visible range, there is an important advantage of using them as EMI shield for high-frequency nanoelectronics: they both can be deposited onto the top of dielectric substrate directly (any shape and size), and therefore can be used not as a part of sandwich structure, but as they are.

EMI shield for high frequency electronics
Design of compact effective shielding layers of microwave radiation is a very important goal for many practical applications. For development of 5G communication systems, it is necessary to solve many problems, related to EM compatibility (EMC) and propose materials perfectly absorbing EM radiation at high frequencies, up to THz range. The subject of this subsection is to show at which thicknesses (number of layers in case of sandwich structures) one may expect high enough EMI shielding efficiency (SE) coursed by absorption mostly. realization. If the substrate has smaller permittivity, the optimal lopt carbonaceous film thickness corresponding to minimal reflection and maximal absorption at normal incidence will decrease as =1+lopt/(20c) [20].
For EM attenuation given in Table 1 the absorption value is about 0.84-0.86, whereas R= 0.02-0.03 and T = 0.11-0.013. In free space, the values of EMI SE will be somewhat higher (11-13 dB).  Fig.9. Example of calculation of optimal number of layers in sandwich structure for particular conductivities of carbonaceous films (0.28ε0c, graphene in this particular case) corresponding to minimal reflection (when radiation comes from the side of 1.44 mm silica substrate) and maximal absorption (>85%). 20  Using the method of conductivity calculations from the measured electromagnetic response of the samples in the THz frequency range (details of the calculation are presented in [4]), it is easy to show that graphene, GrI and GrPyC are characterized by the following conductivity values:

Experimental data and comparative analysis
0.21 ε0c, 0.36ε0c and 0.37ε0c. 21 Theoretically calculated dependences of absorption and transmission of s and p polarized wave in free-standing thin carbonaceous film vs the angle of incidence and conductivity are shown in Figure 11. Analyzing the experimental data obtained in the context of theoretical estimates, we can conclude that the increase of the overall conductance of the thin carbon film or sandwich leads to sharper absorptance angle dependence for s-polarized wave (see also [4]), which can be used in the development of tunable THz passive device.
In this regard, the use of graphene is more favorable in comparison with other types of investigated carbon materials. However, it should be noted that the effects associated with selective transmission of differently polarized waves on the angle of incidence are more pronounced for the films with higher conductivity (therefore for mass production of THz modulator or filter either GC or PyC directly deposited on the top of quartz substrate, or GrPyC which does not need PMMA , could be more appropriate).

Filter, polarizer, modulator, collimator
The different reaction of thin conductive film, including graphene, PyC, GrPyC, GC to react differently to the polarized radiation can be used for design and fabrication different passive devices for THz applications. As it was discussed in section 2.1 for s-polarized radiation the optimal thickness (number of layer) decreases dramatically with the increase of incidence angle, which means suppressing transmittance ability of thin carbon film for s-polarized waves at almost sliding incidence. In contrast, for p-polarized radiations, the thickness of carbon layer required for maximal absorption increases together with increasing incidence angle, that is carbonaceous film/sandwich becomes transparent for EM radiation. This effect can be easily used for producing filters, modulators, collimators and polarizers for THz radiation. The schematic representation of such devices is presented in Figure 12.

EMI shield Robust Design approach: the preliminary phase
With the aim of transferring from lab to industrial scale the application of sandwich structures made of carbon-based thin conductive films separated by polymer slabs as electromagnetic shield, a virtual prototyping approach is considered on the schematic model reported in Figure 13 [21, 22]. method, adopted also for the system considered in this paper is described schematically in Figure   14.

Statement of the problem
For the multilayer shield the considered system's performance is the shielding efficiency SE computed by means of a system's model shown in Figure 13. A FEM based commercial software (Comsol Multiphysics® RF Tool) has been adopted for the required simulations. Fig. 15. Simulated geometry and domains of the device in Figure 13.
By considering the geometry and the domains reported in Figure 15, Planar-Wave condition has been applied at "Port 1" (Electric Field: E1); "Port 2" (Electric Field: E2) is the "Output Port".
Perfect Matched Layer (PML) represents "matched loads condition", necessary to avoid possible reflected incident waves causing simulation errors. The structure is infinitely extended along y-direction: in Figure 15 "Periodic Condition" on upper and lower boundaries are fixed and "field continuity condition" along y-direction is applied. Over the whole structure the following equation was implemented where ko is the propagation vector amplitude in vacuum (empty space), r is the relative electric permittivity of each material, is the electrical conductivity, r is the relative magnetic permeability, ω is the frequency of the incident wave and E is the Electric Field. The post processing extraction of the scattering parameters leads to compute the SE= -S21 2 for the considered configuration.

Design parameters
Three design parameters or factors are considered: the number of replicated "cells" in the structure, as illustrated in Figure 13 (3 in the considered case); isthe electrical sheet conductivity of the carbonaceous layer (since it is a new-generation material its fabrication intrinsically may suffer from large uncertainties); lpmma is the thickness of the PMMA layer. The factor NumArray is a discrete controllable parameter ranging between 1 and 9. In the performed evaluations 3 discrete level (i.e. 1, 5 and 9) have been considered. The factors  and lpmma are continuous and partially controllable parameters for which we have adopted a range between 1/10 and 10 times their nominal values, sampled with a 5 level uniform segmentation. The selection and evaluation of the response variable has been obtained by means of a Matlab® environmental routine developed in order to compute the SE at 30 GHz for a given combination of these three factors by exploiting the FEM model described in the previous section.

Design of Experiments (DoE)
By considering a full factorial approach on the design parameter defined in the previous section, a Matlab® procedure able to represent DoE Scatter Plot after computing of 3x5x5=75 total design was developed. The achieved results are shown in Figure 16. From the simulated data reported as blue circles in Figure 16 it is possible to observe that the PMMA-thickness effect becomes relevant only for higher number of replicated cells. In fact, for NumArray = 1 (one cell) the SE remains constant with respect to thickness variation whereas a weak dependence is detected only for NumArray=9. Moreover, SE exhibits a quadratic-type dependence both on the conductivity of the thin layer and on the PMMA thickness (dashed lines in Figure X4). In Figure 17 the same simulated data are used to derive the main effect of each factor.

Response parameters and RSM
According to the dependences found in the previous phase, the RSM Tool function of Matlab is used to calculate the coefficients of the second order approximating function SE for a fixed value In Table 2 the gamma coefficients of the interpolating functions are reported useful to derive the SE in non-simulated design or to apply optimization procedure considered in the schematic layout of Figure 14. In particular, the Robust design of the system with the procedure proposed in [25] and based on the interpolating functions here obtained is still in progress and will appear in a forthcoming paper.

Conclusion
We discussed the possibility to use various conductive carbon films as building blocks for passive electromagnetic devices, including EMI shielding layer, filters, polarizers, modulators and collimators for microwave and THz radiation. The peculiarities of different carbonaceous films, including the simplicity or difficulty of their synthesis and optical transparency were discussed. We found that for microwave application, any of the investigated carbon films could be used (optimal thickness and number of layers in the sandwich structures were found). In case, when optical transparency is a necessary condition, graphene, GrI and GrPyC are the best candidates, with an advantage for graphene. In case one needs only shielding ability, PyC and GC are good alternative as they can be deposited on the dielectric substrate directly.
The ability of thin conductive film, including graphene, PyC, GrPyC, GC to respond to polarized radiation can be used for the design and fabrication different passive devices for THz applications. A demonstrator of PyC to be used as the sensitive element in the THz polarizer was presented.
The final conclusion is that, depending on the particular application that is targeted, both graphene and other carbonaceous films (GrPyC, GrI, PyC, GC) could be utilized. In case one need a shield for optoelectronic window, the best option is graphene, GrPyC and GrI (but graphene is the best because of the best transparency in visible range). In case of filters, collimators, modulators, where dielectric substrate should be also used, PyC and GC, which can be deposited directly without the delicate PMMA shuttle transfer, could be better than graphene. In both cases, the angles θI and θT are related by the Snell-Descartes law nIsin θI = nT sin θT.            , the "Campania Regional Competence Centre for Food Production" and the "Campania Regional Competence Centre for Food Production". Her research interests concern numerical methods for electromagnetic fields, tolerance analysis and design optimization in power electronics circuits and magnetic components. Since 2002 she is involved in experimental researches on materials and innovative composites for electrical engineering applications at the DIIIE Lab for Electromagnetic Characterization of Materials. The research activity has lead to several