Multi-channel graphene-based perfect absorbers utilizing Tamm plasmon and Fabry-Perot resonances

Maryam Heidary Orojloo; Masoud Jabbari; Ghahraman Solooki Nejad; Foozieh Sohrabi

doi:10.1364/OE.515659

1. Introduction

Surface electromagnetic waves are electromagnetic modes that are excited at the interface of two distinct materials and decay evanescently [1–3]. Tamm plasmon polariton (TPP) is a form of electromagnetic surface wave that was theoretically postulated in 2007 [4] and physically seen in 2008 [5]. Because of the strong light-matter interaction, the Tamm plasmon opens up a new avenue in the plasmonics of interface states in planar multilayer structures [6]. A TPP mode is excited at the interface of a metal with negative permittivity and a one-dimensional photonic crystal (1D-PhC) or a distributed Bragg reflector (DBR) [2,7,8]. Tamm states are sensitive to changes in parameters of the PhC such as periodicity, refractive index, layer order, the insertion of additional layers, and modifying the thickness of the layer next to the metal [9,10]. Tamm plasmon, in contrast to patterned surface plasmon devices, provides a simple planar solution with significant field augmentation at the interface and does not need any phase-matching mechanism for excitation. Unlike surface plasmon polaritons, TPP modes are polarization-independent [11,12] and may be optically stimulated without the use of phase-matching methods such as the application of gratings or prisms [4]. Tamm plasmons are an appealing subject of research with possible practical applications due to the simplicity of producing multilayer thin film stacks, direct optical stimulation, and high-Q modes [13].

In recent years, TPPs have attracted the great attention of researchers. Different optical devices such as lasers [14–16], sensors [17–19], photodetectors [20,21], and absorbers [22–24] have been reported based on Tamm plasmons. Absorbers are universal devices that are utilized in different applications such as filtering, sensing, thermal emission manipulation, solar energy harvesting, and thermo-photovoltaic systems. Gong et al. reported an absorber utilizing Tamm modes in a 1D-PhC, consisting of TiO₂ and SiO₂ layers [25]. TPPs are excited due to the use of a thin metal layer, resulting in a high absorption of 99.1%. The resonance wavelength can be tuned from 590 nm to 1550 nm by adjusting the structure's geometrical variables. Lu et al. checked the performance of a wide-angle perfect absorber by simulation and experimentally [26]. The absorber consists of a metal layer and a 1D-PhC, built of layered hyperbolic metamaterials and dielectrics. The absorber has an absorption peak of 91% due to TPP resonance for incident angle variations from 0° to 45°. As another example, Li et al. presented an absorber that operates in the wavelength range of 1510-1690 nm [27]. They used a metal layer adjacent to a PhC to stimulate Tamm plasmons. Employing graphene instead of the metallic layer paves the way to excite Tamm plasmons in the terahertz (THz) region. Graphene, as a two-dimensional material, can act as a single-layer metallic layer [28]. Lu et al. proposed a dual-band perfect absorber based on a graphene sheet and an all-dielectric PhC, working in the wavelength range of 200-400 µm [29]. Another graphene-based absorber was designed by Wang et al. to operate in the frequency range of 0.85-1.15 THz [30]. The absorber consists of a graphene sheet, a dielectric layer, and an all-dielectric PhC. The absorber can attain an absorption of 98.6% by changing the geometrical parameters of the structure. As expressed, a metallic layer adjacent to a PhC or a DBR must be used to stimulate TPPs. Conventional metals such as silver and gold can be used to excite TPPs at visible and infrared wavelengths. However, graphene is a suitable material to excite TPPs in the THz region. According to our survey, previously reported works used all-dielectric PhC, and the effects of using dielectric-metal PhC structures on the absorption behavior of TPP-based absorbers have not been investigated. Therefore, in this paper, we investigate the absorption behavior of absorbers comprised of a graphene sheet and dielectric-metal PhC in the range of 1-2 THz.

The rest of the paper is organized as follows: In Section 2, the basic PhC and the theoretical relationships and formulation of the TMM are presented. In Section 3, the simulation results are given. The influences of different parameters affecting the absorption spectrum of the structure are investigated in this section. The conclusion is stated in Section 4. Additionally, some additional descriptions are provided in the appendix section.

2. Principles of basic suspended graphene waveguide

Figure 1 shows the schematic of the proposed structure. It consists of a graphene sheet on top, a dielectric layer called the spacer layer, and N periods of alternating layers of A and B, all located on a dielectric substrate. The material and thickness of layers A are shown by n_A and D_A, respectively. Also, n_B and D_B refer to the material and thickness of layers B. The period of the PhC is a = D_A + D_B. The thickness and material of the spacer layer are denoted by D_C and n_C. The materials of the spacer and substrate layers are assumed to be the same. The incident light with a wavenumber of k₀ and an incident angle of θ₀ irradiates the surface of the absorber. The thickness of N periods of the PhC is equal to N × a.

Fig. 1. Schematic of the studied structure, consisting of a graphene sheet, a spacer layer, and a PhC, located on a dielectric substrate.

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The graphene sheet is modeled based on its surface conductivity (σ = σ_inter + σ_intra) using the Kubo formula as follows [30]:

(1)$${\sigma _{\textrm{inter}}} = \frac{{i{e^2}}}{{4\pi \hbar }}\ln \left( {\frac{{2|{{\mu_c}} |- ({\omega + i{\tau^{ - 1}}} )\hbar }}{{2|{{\mu_c}} |+ ({\omega + i{\tau^{ - 1}}} )\hbar }}} \right)$$

(2)$${\sigma _{\textrm{intra}}} = \frac{{i{e^2}{k_B}T}}{{\pi {\hbar ^2}({\omega + i{\tau^{ - 1}}} )}}\left[ {\frac{{{\mu_c}}}{{{k_B}T}} + 2\ln \left( {1 + {e^{ - \frac{{{\mu_c}}}{{{k_B}T}}}}} \right)} \right]$$

where e, ħ, and k_B denote the electron charge, reduced Planck’s constant, and Boltzmann's constant, respectively. Additionally, the parameters μ_c, ω = 2πf, T, and τ = (2Γ)^-1 are the chemical potential, angular frequency, temperature, and relaxation time. The parameter Γ refers to the scattering rate of graphene [30]. Researchers recently reported that the chemical potential can attain high values up to 1.17 eV [31]. A graphene sheet that is grown by the chemical vapor deposition method shows carrier mobility of 1000 cm²/V.s [32], and this value can increase up to 230000 cm²/V.s for a suspended exfoliated graphene sheet [33]. The carrier density of graphene is controlled by applying chemical doping or electrostatic doping [34]. The carrier density changes result in variations in the chemical potential and surface conductivity of graphene.

When light impinges the graphene sheet, two amplitude reflection coefficients in the spacer layer appear as seen in Fig. 2, called r_g and r_PhC. The parameter r_g denotes the reflection coefficient for light propagating in the -z direction, while the parameter r_PhC is the reflection coefficient for light propagating in the + z direction. To excite Tamm plasmons, the amplitude matching condition must be satisfied. Utilizing the TMM, Eq. (3) is achieved [4,29].

(3)$$K\left( \begin{array}{l} 1\\ {r_g} \end{array} \right) = \left( \begin{array}{l} \textrm{exp} ({i\phi } )\textrm{ 0}\\ \textrm{ }0\textrm{ exp}({ - i\phi } )\end{array} \right)\left( \begin{array}{l} {r_{PhC}}\\ 1 \end{array} \right)$$

where K is a constant, and ϕ is the phase change due to the light propagation in the spacer layer. After removing the coefficient K and performing several mathematical simplifications, the excitation condition of Tamm plasmons is obtained as follows [29]:

(4)$${r_g}{r_{PhC}}\textrm{exp} (2i\phi ) \approx 1$$

Fig. 2. Schematic of reflection waves in the spacer layer.

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By knowing the transmission (T) and reflection (R) spectra of the structure, the absorption (A) spectrum is achieved as follows [35]:

(5)$$A(f )\textrm{ } = \textrm{ }1\textrm{ }-{-}\textrm{ }R(f )\textrm{ }-{-}\textrm{ }T(f )$$

The reflection and transmission coefficients of the proposed structure can be calculated using the modified TMM. For a p-polarized incident wave, the transmission characteristics of light through the interface between the lth layer and the mth layer with graphene can be calculated by matrix D_lm, which can be expressed as follows [29]:

(6)$${D_{lm}} = \frac{1}{2}\left[ \begin{array}{l} 1 + {\eta_p} + {\xi_p}\textrm{ }1 - {\eta_p} - {\xi_p}\\ 1 - {\eta_p} + {\xi_p}\textrm{ 1 + }{\eta_p} - {\xi_p} \end{array} \right]$$

where

(7)$${\eta _p} = \frac{{{\varepsilon _l}{k_{mz}}}}{{{\varepsilon _m}{k_{lz}}}}$$

(8)$${\xi _p} = \frac{{\sigma {k_{mz}}}}{{{\varepsilon _0}{\varepsilon _m}\omega }}$$

(9)$${k_{lz}} = {k_0}\sqrt {{\varepsilon _l} - {\varepsilon _0}{{\sin }^2}{\theta _0}}$$

(10)$${k_{mz}} = {k_0}\sqrt {{\varepsilon _m} - {\varepsilon _0}{{\sin }^2}{\theta _0}}$$

Here, ɛ₀ and σ are the permittivity of the free space and the surface conductivity of graphene, respectively. Additionally, ɛ_l and ɛ_m are the permittivity of the lth and mth layers, respectively. For an s-polarized wave, the D_lm matrix is as follows [29]:

(11)$${D_{lm}} = \frac{1}{2}\left[ \begin{array}{l} 1 + {\eta_s} + {\xi_s}\textrm{ }1 - {\eta_s} + \xi \\ 1 - {\eta_s} - \xi \textrm{ 1 + }{\eta_s} - \xi \end{array} \right]$$

where

(12)$${\eta _s} = \frac{{{k_{mz}}}}{{{k_{lz}}}}$$

(13)$${\xi _s} = \frac{{\sigma {\mu _0}\omega }}{{{k_{lz}}}}$$

Here, µ₀ is the permeability of the free space. It is worth noting that a p-polarized light or transverse magnetic (TM) mode refers to a light wave whose electric field vector is in the incidence plane, whereas an s-polarized light or transverse electric (TE) mode refers to a light wave whose electric field vector is normal to the incidence plane [36,37]. The transmission and reflection spectra of the proposed structure are achieved using the D_lm matrices for p- and s-polarized waves. As a result, the absorption spectrum is found.

3. Simulations and results

The proposed structure is simulated using the TMM method, presented by the commercial Lumerical software. The software provides a user-friendly and simple programming platform for studying the characteristics of multi-layer structures. The values of the Kubo parameters for modeling the graphene sheet are T = 300° K, Γ = 0.11 meV, and µ_c = 0.7 eV. The materials of layers A and B are assumed to be dielectrics with non-dispersive refractive indices of n_A = 1.9 (MgF₂) and n_B = 3.53 (AlGaAs), respectively. The thicknesses of the spacer layer, layers A, and layers B are D_C = 50 µm, D_A = 200 µm, and D_B = 50 µm, respectively. The number of periods is N = 8. Also, the incident light irradiates perpendicular to the surface of the structure (θ₀ = 0°). Figure 3 illustrates the reflection, transmission, and absorption spectra of the proposed structure. The number of three perfect photonic bandgaps can be seen in this figure, where transmission is zero. The absorption spectrum of the structure is achieved using Eq. (5). The structure has four absorption channels at f₁ = 1.042 THz, f₂ = 1.315 THz, f₃ = 1.602 THz, and f₄ = 1.915 THz in the frequency range of 1-2 THz. The corresponding absorption values are A₁ = 50.72%, A₂ = 45.40%, A₃ = 22.16%, and A₄ = 23.21%, respectively. The origin of absorption at these frequencies is attributed to Tamm plasmon resonances. It has been expressed that these absorption peaks are because of the combination of the graphene sheet and the PhC. The graphene sheet acts like a metal, and the PhC serves as a distributed Bragg reflector (DBR). Figure 4 reveals the effect of the graphene sheet on the spectral behavior of the structure. When the graphene sheet is removed, Tamm plasmons are not excited. Therefore, the absorption is almost zero.

Fig. 3. Reflection, transmission, and absorption spectra of the structure for n_A = 1.9, n_B = 3.53, µ_c = 0.7 eV, D_A = 200 µm, D_B = D_C = 50 µm.

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Fig. 4. Comparing the (a) Reflection, (b) transmission, and (c) absorption spectra of the structure for the presence and absence of the graphene sheet for n_A = 1.9, n_B = 3.53, D_A = 200 µm, D_B = D_C = 50 µm.

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As seen in Fig. 3, the absorption values are relatively low for all-dielectric PhC. Therefore, the effect of utilizing dielectric-metal PhC is investigated. For this purpose, the material of layer B is changed to silver (Ag) metal. The permittivity of Ag is calculated using the Drude-Lorentz model as follows [38]:

(14)$$\varepsilon (\omega )= {\varepsilon _{r,\infty }} + \sum\limits_{k = 0}^K {\frac{{{f_k}\omega _p^2}}{{\omega _k^2 - {\omega ^2} + j\omega {\Gamma _k}}}}$$

where ɛ_r_,∞ is the dielectric constant at infinite frequencies and ω_p is the plasma frequency. Additionally, the parameters ω_k, f_k, and Γ_k are the resonance frequency, strength, and damping frequency of the kth oscillator, respectively. The values of the parameters used in the Drude-Lorentz model for Ag metal are given in the appendix section.

Figure 5 presents the absorption spectra of the structure with a dielectric-metal PhC for two different dielectric materials. The thickness of the metallic layers is 10 nm. For the PhC with n_A = 1.9, three absorption channels are seen at frequencies of f₁ = 1.302 THz, f₂ = 1.619 THz, and f₃ = 1.927 THz with absorption peaks of A₁ = 87.44%, A₂ = 66.49%, and A₃ = 56.52%. Compared to all-dielectric PhC, the dielectric-metal PhC represents higher absorption values. However, the free spectral range (FSR) is increased, which results in a lower number of absorption channels in the range of 1-2 THz. The increment of the refractive index of the dielectric layer (n_A) from 1.9 to 3.53 increases the absorption peaks significantly so that in the range of 1 to 2 THz, five absorption channels with absorption values of A₁ = 94.94%, A₂ = 99.23%, A₃ = 83.06%, A₄ = 60.64%, and A₅ = 61.03% are observed at f₁ = 1.153 THz, f₂ = 1.351 THz, f₃ = 1.54 THz, f₄ = 1.714 THz, and f₅ = 1.881 THz, respectively. Therefore, it can be concluded that the higher the refractive index of the dielectric layers, the higher the number of absorption channels. This phenomenon can be attributed to the Fabry-Perot effect in the sequence of graphene, spacer, dielectric, and metal layers. When the refractive index of the medium between the partial mirrors of a Fabry-Perot interferometer increases, the number of resonance frequencies increases, too. The detailed discussion is given in the appendix section.

Fig. 5. Absorption spectra of the structure with a dielectric-metal_(Ag) PhC for two different values of n_A. The other parameters are µ_c = 0.7 eV, D_A = 200 µm, D_B = 10 nm, and D_C = 50 µm.

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To determine the contribution of each optical resonance on the absorption behavior of the structure, four different structures are investigated. The arrangement of layers in the structures is as follows:

Case 1: Spacer-[dielectric_(1.9)-dielectric_(3.53)]⁸-substrate

Case 2: Spacer-[dielectric_(1.9)-metal_(Ag)]⁸-substrate

Case 3: Graphene-spacer-[dielectric_(1.9)-dielectric_(3.53)]⁸-substrate

Case 4: Graphene-spacer-[dielectric_(1.9)-metal_(Ag)]⁸-substrate

The expression []⁸ indicates the materials and the number of periods of the PhC. As seen in Fig. 6, the all-dielectric PhC in the absence of the graphene sheet has almost zero absorption (Case 1). Replacing the dielectric material with a refractive index of 3.53 with Ag metal causes weak resonances in the absorption spectrum (Case 2). In Case 3, a graphene sheet is added to the structure with an all-dielectric PhC. In this case, Tamm plasmons are excited, and relatively strong resonances are observed in the absorption spectrum. The arrangement of the layers in Case 4 is similar to Case 3, with the difference that the dielectric with a refractive index of 3.53 is replaced by Ag metal. In this case, strong resonances are seen in the absorption spectrum, resulting from the coupling of the Tamm plasmon and Fabry-Perot resonances.

Fig. 6. Absorption spectra for different cases of the proposed structure.

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To have a physical understanding of how electromagnetic waves are absorbed inside the structure, the distributions of the magnitude of the electric and magnetic fields along the structure are plotted. Figure 7 shows the fields’ distributions for the structure with a dielectric_(3.53)-metal_(Ag) PhC. The other parameters are n_C = 1.45, µ_c = 0.7 eV, D_A = 200 µm, D_B = 10 nm, and D_C = 50 µm. As seen, the electric and magnetic fields are mainly concentrated at a distance of 250 µm from the beginning of the structure. When the incident waves impinge the Ag metal layer of the first period of the PhC, the intensity of the electric and magnetic fields is severely reduced. In other words, the Ag metal of the first period of the PhC acts as a semi-mirror.

Fig. 7. Distribution of the magnitude of the (a) electric field and the (b) magnetic field along the structure. The PhC is a dielectric_(3.53)-metal_(Ag) type. The other parameters are n_C = 1.45, µ_c = 0.7 eV, D_A = 200 µm, D_B = 10 nm, and D_C = 50 µm.

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As proved previously, the graphene sheet has a remarkable effect on the absorption behavior of the structure. The refractive index of graphene is a function of its chemical potential. Therefore, changing the chemical potential of the graphene sheet may alter the spectral response of the structure. The absorption spectra of the structure for chemical potential values equal to µ_c = 0.6 eV, 0.7 eV, and 0.8 eV are drawn in Fig. 8. It can be seen that with an increase in the chemical potential, the resonance frequencies have redshifts in the higher frequencies. Meanwhile, the absorption values increase, indicating the graphene chemical potential is more effective at relatively high frequencies.

Fig. 8. Absorption spectrum of the structure with a dielectric_(3.53)-metal_(Ag) PhC for different values of graphene chemical potential. The other parameters are n_C = 1.45, D_A = 200 µm, D_B = 10 nm, and D_C = 50 µm.

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Since the incident light penetrates the spacer layer before the PhC, a change in the refractive index of the spacer layer will alter the absorption spectrum. Figure 9 compares the absorption spectra of the structure for n_C = 1.9 and n_C = 3.53. It is clear that the absorption values except for one resonance are higher for n_C = 1.9 compared to those in the structure with n_C = 3.53. With an increase in n_C, the absorption value and the FSR decrease.

Fig. 9. Absorption spectra of the structure with a dielectric_(3.53)-metal_(Ag) PhC for two different values of n_C. The other parameters are µ_c = 0.7 eV, D_A = 200 µm, D_B = 10 nm, and D_C = 50 µm.

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In the following, the effect of changes in the thicknesses of the PhC on the absorption spectrum is analyzed. Figure 10(a) demonstrates the absorption spectrum for variations in the thickness of the dielectric layers from 50 µm to 300 µm. In this case, the thickness of the metallic layers is assumed to be 10 nm. As the thickness of the dielectric layers increases, the number of resonances increases, too. For instance, the structure resonates at frequencies of f₁ = 1.414 THz and f₂ = 1.793 THz for D_A = 70 µm, while the structure shows seven resonances at frequencies of f₁ = 1.128 THz, f₂ = 1.273 THz, f₃ = 1.415 THz, f₄ = 1.553 THz, f₅ = 1.686 THz, f₆ = 1.81 THz, and f₇ = 1.94 THz for D_A = 280 µm. Unlike the thickness changes of the dielectric layers, the number of resonances is independent of the thickness of the metallic layers. This issue can be seen in Fig. 10(b). In addition, the thinner the metallic layer, the wider the absorption bandwidth, and at the same time, the higher the absorption value.

Fig. 10. Absorption spectrum of the structure with a dielectric_(3.53)-metal_(Ag) PhC as a function of the thickness variations of (a) dielectric layers (D_A) and (b) metal layers (D_B).

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Finally, the effect of the incident angle on the absorption spectrum of the presented structure is investigated. Under normal incidents, the response of the structure for p-polarized and s-polarized waves is the same because of the symmetric of the structure. However, the structure shows different absorption spectra for p- and s-polarization of oblique incident light. The absorption spectrum of the structure for p- and s-polarized oblique incidents up to 80° is displayed in Fig. 11.

Fig. 11. Absorption spectrum of the structure with a dielectric_(3.53)-metal_(Ag) PhC under oblique radiation from 0° to 80° for (a) p-polarized light and (b) s-polarized light.

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4. Conclusion

In this paper, the absorption behavior of a structure composed of a graphene sheet located adjacent to a PhC was comprehensively studied. The structure serves as a tunable multi-channel absorber in the frequency range of 1-2 THz, taking advantage of the Tamm plasmon and Fabry-Perot resonances. Absorption values and resonance frequencies can be controlled by changing the material and thickness of the layers as well as the chemical potential of the graphene sheet. It was shown that using dielectric-metal PhC has a better absorption response compared to the all-dielectric ones. Furthermore, the effect of polarization and the angle of the incident light on the structure’s absorption spectrum was investigated. The proposed structure has high potential in promoting sensing and filtering applications.

Appendix

A. Drude-Lorentz model of silver metal

The Drude-Lorentz model is an advanced type of the Drude model because it incorporates the bound-electron effects into the initial Drude model. The validity range of the Drude model can be developed by including the Lorentzian term. The Lorentz-Drude model utilizes K-damped harmonic oscillators to characterize small resonances that exist in the spectral response of metals. Table 1 represents the parameter values of this model for silver metal, taken from Ref. [39]. The real and imaginary parts of the permittivity of Ag metal for 1-2 THz are plotted in Fig. 12.

Fig. 12. (a) Real and (b) imaginary parts of the permittivity of silver (Ag) in the frequency range of 1-2 THz.

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Table 1. Parameters of the Drude-Lorentz model for silver metal

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B. Distribution of the electric and magnetic fields

Figure 13 shows the distributions of the magnitude of the electric and magnetic fields along the structure at one non-resonance frequency and five resonance frequencies. The structure’s parameters are n_C = 1.45, n_A = 3.53, n_B = n_Ag, D_C = 50 µm, D_A = 200 µm, D_B = 10 nm, and µ_c = 0.7 eV. Comparing the amplitude of the fields at the resonance and non-resonance frequencies implies a high field intensity at the resonance frequencies, leading to a high absorption of the incident light at these frequencies. Furthermore, the first period of the PhC is mainly responsible for the resonance mechanism and absorption of the incident light. It can be seen that the incident light is completely absorbed after the third period (z < 600 µm).

Fig. 13. Distributions of the magnitude of the (a) electric field and (b) magnetic field along the structure

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C. Fabry-Perot resonator

A Fabry–Perot resonator consists of two semi-mirrors at a distance of D from each other with a dielectric medium (cavity) between them. An electromagnetic wave is reflected repeatedly within a Fabry-Perot resonator by the semi-mirrors. The reflected waves’ interference causes a periodic fluctuation in the transmitted optical power as a function of wavelength. When the reflections are in phase, the wave interferes constructively, and as a result, the highest transmission value is achieved. When the reflections are out-of-phase, the wave interacts destructively, and the lowest transmission value is obtained. The resonance frequencies of a Fabry-Perot resonator can be calculated by the following equation [40]:

(15)$${f_{res}} = \frac{{mc}}{{2{n_{eff}}D}},\textrm{ }m\textrm{ } = \textrm{ }1,\textrm{ }2,\textrm{ }3,\textrm{ }\ldots $$

where n_eff is the effective refractive index of the cavity, c is the speed of light in free space, and m is an integer. In a dielectric-metal PhC, it can be considered that several Fabry-Perot resonators are stacked together. Due to the periodicity of the dielectric-metal PhC, the transmitted light is absorbed. In the proposed structure, the graphene sheet and the first Ag layer of the PhC can be assumed as the semi-mirrors. The schematic of the Fabry-Perot resonator is shown in Fig. 14.

Fig. 14. Schematic of the Fabry-Perot resonator.

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For this resonator, the n_eff is estimated using Eq. (16). The length of the cavity is equal to D = D_spacer + D_dielectric.

(16)$${n_{eff}} = \frac{{({{n_{spacer}} \times {D_{spacer}}} )+ ({{n_{dielectric}} \times {D_{dielectric}}} )}}{{{D_{spacer}} + {D_{dielectric}}}}$$

For n_spacer = 1.45, n_dielectric = 1.9, D_spacer = 50 µm, and D_dielectric = 200 µm, the resonance frequencies in the range of 1-2 THz are f₁ = 1.3250 THz, f₂ = 1.6563 THz, and f₃ = 1.9875 THz. By varying the n_dielectric from 1.9 to 3.53, Eq. (15) provides five resonance frequencies including f₁ = 1.1552 THz, f₂ = 1.3478 THz, f₃ = 1.5403 THz, f₄ = 1.7329 THz, and f₅ = 1.9254 THz. These resonance frequencies in both cases are relatively consistent with the resonance frequencies obtained by the TMM. Table 2 comprises the resonance frequencies obtained from the TMM and the Fabry-Perot resonator estimation. This comparison is also shown in Fig. 15.

Fig. 15. Resonance frequencies obtained by the simulation and calculation methods as a function of the resonance number for two values of n_A.

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Table 2. Comparison between resonance frequencies obtained by the TMM and Fabry-Perot model

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Disclosures

The authors declare no conflicts of interest.

Data availability

The calculated results during the current study are available from the corresponding author on reasonable request.

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k	1	2	3	4	5	6
ω_k	0	0.816	4.481	8.185	9.083	20.29
f_k	0.845	0.065	0.124	0.011	0.840	4.646
Γ_k	0.048	3.886	0.452	0.065	0.916	2.419

k	1	2	3	4	5	6
ω_k	0	0.816	4.481	8.185	9.083	20.29
f_k	0.845	0.065	0.124	0.011	0.840	4.646
Γ_k	0.048	3.886	0.452	0.065	0.916	2.419

Multi-channel graphene-based perfect absorbers utilizing Tamm plasmon and Fabry-Perot resonances

Abstract

1. Introduction

2. Principles of basic suspended graphene waveguide

3. Simulations and results

4. Conclusion

Appendix

A. Drude-Lorentz model of silver metal

B. Distribution of the electric and magnetic fields

C. Fabry-Perot resonator

Disclosures

Data availability

References

Data availability

Cited By

Figures (15)

Tables (2)

Equations (16)

Optics Express

n_dielectric	Method	Res. 1 (THz)	Res. 2 (THz)	Res. 3 (THz)	Res. 4 (THz)	Res. 5 (THz)
1.9	TMM	1.302	1.619	1.927	-	-
1.9	Fabry-Perot estimation	1.3250	1.6563	1.9875	-	-
3.53	TMM	1.153	1.351	1.540	1.714	1.881
3.53	Fabry-Perot estimation	1.1552	1.3478	1.5403	1.7329	1.9254

Ghahraman Solooki Nejad	https://orcid.org/0000-0002-5521-0190
Foozieh Sohrabi	https://orcid.org/0000-0003-4105-6839