1. Introduction

Mid-infrared electromagnetic (EM) waves, with wavelengths typically ranging from 3 to 30 μm, have attracted intensive research interest because of their wide application in the fields of communication, biomedicine, spectroscopy, homeland security [1, 2], etc. It would be highly desired to design an optical waveguide with ultra-long propagation length L_prop and ultra-deep sub-wavelength confinement, because such a device can be used for a basic component to transmit mid-infrared EM waves in certain practical applications, such as optical modulation [3], sensing [4], filtering [5], and near-field imaging [6].

Surface plasmon polartions (SPPs), which are a collective excitation of electrons along a metal-dielectric interface, have relatively low propagation loss as well as deep sub-wavelength confinement from the visible to infrared range [7]. Thus, the SPP-based waveguide is a promising candidate for future application to guide EM waves in this region. Noble metals are typically regarded as the most suitable SPP materials currently available [8], and various noble-metal-based SPP waveguides have been proposed [9–16]. Among them, the long-range SPP (LRSPP) waveguide [13–16], which is composed of a thin metal stripes embedded in infinite homogeneous background dielectric material, can be employed to guide an EM field over a long propagation distance (~10⁴–10⁶ μm). However, these types of LRSPP waveguide suffer from poor field confinement (~10⁻¹–10 A₀, where A₀ is the diffraction limited mode area), which renders them unsuitable for compact integration. Recently, a type of metal-based LRSPP hybrid waveguide [17–21], which is composed of two identical low-index inner dielectric layers and two identical high-index outer dielectric layers symmetrically placed on both sides of a thin metal film, has attracted strong research interest. This attention is because of this waveguide’s ability to achieve deep sub-wavelength confinement (~10⁻²–10⁻¹ A₀) with comparable L_prop (~10⁴–10⁵ μm), as opposed to the conventional LRSPP waveguide. However, these waveguides are only suitable for application from the near-infrared to visible waveband, and are especially focused on the telecommunication wavelength of λ = 1.55 µm. In the mid-infrared range, SPPs have very weak confinement on the metal surface [22,23], rendering this device unsuitable for use as an SPP waveguide in this waveband.

Graphene, a single-layer combined carbon atom sheet [24], is a promising candidate for SPP wave guiding in the mid-infrared range [25,26]^{, and} various graphene-based SPP waveguides have been investigated [22,23,25–28]· Especially, the monolayer and multilayer graphene nanoribbons waveguides have been widely researched and been applied to many fields in recent year [29–35]. Compared with the metal-based SPP waveguide design, the graphene-based SPP waveguide exhibits extremely strong mode confinement, low propagation loss, and tunable electromagnetic properties [36]. Nonetheless, in the mid-infrared waveband, the graphene plasmon also suffers from high absorption loss because the majority of the light energy is located in the graphene. Inevitably, the L_prop of this type of waveguide remains relatively small (~10⁰ μm). Thus, the key challenge to the graphene-based SPP waveguide is determining a means of dramatically improving the L_prop while maintaining the existing (or a similar) degree of confinement.

Combining the unique optical characteristics of graphene with the analogous principles of the metal-based LRSPP hybrid waveguide, in this paper, we propose a graphene-based LRSPP hybrid waveguide, which is composed of two identical outer graphene nanoribbons and two identical inner silica layers symmetrically placed on both sides of a silicon layer. Based on the coupled-mode perturbation theory [37], we demonstrate that the LRSPP and SRSPP modes originate from the coupling of the same modes of the two graphene nanoribbons. Moreover, the LRSPP mode of this waveguide is capable of achieving an ultra-long L_prop (~10 μm), which is the largest L_prop of all the reported graphene-based waveguides to date (to the best of our knowledge). Further, for this waveguide, an ultra-deep sub-wavelength confinement (~10⁻⁷ A₀) similar to that of other conventional graphene-based hybrid waveguides is maintained [28, 29].

2. Theoretical model

Figure 1 is a schematic of the proposed graphene-based LRSPP hybrid waveguide, where, as noted above, two identical outer graphene nanoribbons and two identical inner silica layers are symmetrically placed on both sides of a thin silicon layer. The thicknesses of the silica and silicon layers are labeled as t and d, respectively, and the width of the waveguide is labeled as W. The permittivities of the silica and silicon are ε₁ and ε₂, respectively. Further, the permittivity of graphene can be calculated using ε_g = 2.5 + iσ_g/(ε₀ωΔ), where σ_g is the conductivity of graphene, Δ is the thickness of the given graphene specimen, and ω is the angular frequency of the incident light. Note that σ_g can be calculated from the Kubo formula [34, 38, 39], which depends on the Fermi energy E_f, the ambient temperature T, ω, and the electron relaxation time τ. The latter is related to the electron mobility µ and the Fermi velocity v_F by τ = µE_f /(ev_F²). In our calculations, we set the parameters as T = 300 K, µ = 1.0 m²/(Vs), Δ = 0.5 nm, ε₁ = 2.09, and ε₂ = 11.9, and the incident wavelength in vacuum is λ₀ = 10 µm.

 

Fig. 1 Schematic diagram of the proposed graphene-based LRSPP hybrid waveguide. (a) Cross section of the waveguide. (b) 3D structure of the waveguide.

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It is well known that two types of EM field modes can be simultaneously propagated along this waveguide, including the graphene SPP mode (GSPPM) and the dielectric waveguide mode (DWM). Mutual coupling of the EM fields can occur between any two modes, including GSPPM-GSPPM, DWM-DWM, and GSPPM-DWM. As the coupling strength of the DWM-DWM configuration is significantly smaller than that of GSPPM-GSPPM, DWM-DWM coupling can be ignored. Further, the GSPPM-DWM coupling can also be ignored, because of the effective index mismatching [18]. Thus, the operating pattern of the proposed waveguide is equivalent to GSPPM-GSPPM coupling, and we can study the physical mechanism of the waveguide by investigating the GSPPM-GSPPM coupling characteristics.

Normally, both the graphene nanoribbons can support several SPP modes. To avoid the effective index mismatching mentioned above, we consider the coupling of the same GSPPMs only, and focus on the coupling of fundamental GSPPMs in particular, because they exhibit the maximum coupling strength. According to the coupled mode perturbation theory [37], in the perturbation approximation, the EM-field arbitrary component of this waveguide, ${\Phi }_i(x,y,z)$, can be expressed as a linear superposition of two GSPPMs

We also employ the finite-difference time-domain (FDTD) method by the commercial software of the Lumerical FDTD Solutions to investigate the propagation properties of the proposed waveguide. Due to FDTD is a time-domain technique, the EM fields are solved as a function of time. Based on the time-domain function of the EM fields, the Lumerical FDTD Solutions is used to calculate the EM fields as a frequency-domain function by performing Fourier transforms. The perfectly matched absorbing boundary condition and the non-uniform mesh are applied at the simulation space. The mesh sizes inside the two graphene nanoribbons along the x, y, and z axes are set as 0.05, 0.05, and 5 nm, respectively, and the mesh size gradually increases outside the two graphene nanoribbons. The “mode source” of the Lumerical FDTD Solution, which can solve all EM field modes along the waveguide by mode solver, is used to analyze the characteristics of waveguide modes, including the EM field patterns, the graphene charge distributions, the effective index, and the propagation length. For our proposed waveguide, the TM polarization modes can be obtained by the mode solver of the “mode source”.

3. LRSPP hybrid mode in proposed waveguide

Combining Eq. (2) with the numerical simulation by FDTD, we investigate the mode characteristics of the proposed waveguide. First, we study the mode features of the fundamental SCM and ASCM by varying d. Figure 2(a) shows the effective index n_eff as a function of d for the two modes, which can be expressed as n_eff = n_r + in_i, where n_r and n_i correspond to the real and imaginary parts of n_eff, respectively. β and n_r exhibit similar trend variations for β = n_rβ₀, where β₀ is the propagation constant of incident light in vacuum. With increasing d, energy exchange between the two graphene nanoribbons becomes more difficult. Thus, K_c decreases. As β_s = β + K_c and β_a = β – K_c, β_s and (n_r)_s decrease while β_a and (n_r)_a increase, where (n_r)_s and (n_r)_a correspond to the n_r value of the SCM and ASCM, respectively. When d becomes quite large, K_c tends to 0. Note that β_s ≈β_a ≈(β)_∞ and (n_r)_s ≈(n_r)_a ≈(n_r)_∞, where (β)_∞ and (n_r)_∞ represent the values of β and n_r as d → ∞, respectively. However, when d is very small, (n_r)_s > (n_r)_a, and (λ_SPP)_s < (λ_SPP)_a, where (λ_SPP)_s and (λ_SPP)_a represent the SPP wavelengths of the SCM and ASCM, respectively. For the fundamental SCM and ASCM, when t = 4 nm, W = 100 nm, E_f = 0.8 eV, and d = 8 nm, we can obtain (n_r)_s = 79.6, and (n_r)_a = 21.6. Thus, (λ_SPP)_s = 125.6 nm, and (λ_SPP)_a = 462.9 nm. Figure 2(e) shows that the fundamental SCM completes 10 harmonic oscillations with a propagation distance of 1256 nm, while Fig. 2(f) shows that the fundamental ASCM completes two harmonic oscillations with a propagation distance of 926 nm.

 

Fig. 2 (a) Values of n_eff as functions of d for both SRSPP and LRSPP fundamental modes. (b) Values of L_prop as functions of d for SRSPP, LRSPP, and NHSPP fundamental modes. The insert shows the schematic diagram of the normal hybrid waveguide. (c, d) |E| field distributions of SRSPP and the LRSPP fundamental modes in waveguide cross section, respectively. (e, f) |E| field distributions of SRSPP and LRSPP fundamental modes on waveguide surface, respectively.

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On the other hand, (n_i)_s < (n_i)_a as d > 20 nm. With decreasing d, (n_i)_a first increases gradually and then decreases sharply as d < 30 nm. In particular, when d < 20nm, (n_i)_{s >} (n_i)_a . For instance, for d = 8nm, (n_i)_s = 0.362 and (n_i)_a = 0.146. In that case, the loss of the fundamental SCM is significantly larger than that of the fundamental ASCM, owing to the fact that the loss of an optical waveguide is given by l_m [dB/µm] = –8.86n_i k₀ [40]. Therefore, from Figs. 2(e) and 2(f), we can determine that the electric |E| field distribution of the SCM is attenuated significantly along the propagation direction of the SPP wave, while that of the ASCM exhibits no obvious attenuation.

The propagation length can be expressed as L_prop = 1/[Im(n_eff)k₀] = λ₀/2πn_i [23, 29]. Figure 2(b) shows (L_prop)_s > (L_prop)_a as d > 20nm, where (L_prop)_s and (L_prop)_a represent the propagation lengths of the fundamental SCM and ASCM, respectively. However, (L_prop)_a increases dramatically as d < 20 nm, and is significantly larger than (L_prop)_s. Therefore, we can define SCM and ASCM as the SRSPP and LRSPP modes, respectively. These definitions are exactly opposite to those of the metal-based LRSPP hybrid waveguide [17–21]. To compare the propagation performance of our proposed waveguide, we also investigate the propagation properties of a normal graphene-based hybrid waveguide, which is composed of a silica layer sandwiched between a graphene layer and a silicon layer, as shown in the insert of Fig. 2(b). The parameters of this waveguide are same with that of the LRSPP hybrid waveguide. Figure 2(b) show that, with decreasing d, the propagation length of the normal hybrid SPP (NHSPP) fundamental mode (L_prop)_n first decrease gradually and then increase rapidly as d < 12 nm. This attributes the fact that the SPP confinement decrease with d. Nonetheless, (L_prop)_n is significantly smaller than (L_prop)_a as d < 20 nm. This indicates that the increase of (L_prop)_a not only comes from the decrease of the SPP confinement, but also mostly comes from the increase of the coupling of the two graphene nanoribbons.

Next, we study the physical mechanism behind the difference between the LRSPP and SRSPP modes. For simplicity, we further define the SRSPP and LRSPP fundamental mode as modes 1' and 1, respectively. It is possible to differentiate between these modes by analyzing their respective E_y field distributions [29]. Figures 3(a), 3(b), and 3(e) show that the E_y field distribution of mode 1 is anti-symmetric with respect to the x-z plane. Conversely, Figs. 3(c), 3(d), and 3(f) show that the E_y field distribution of mode 1' is symmetric with respect to the x-z plane. Further, using the EM-field boundary condition, $\overrightarrow{n}\cdot ({\overrightarrow{D}}_2-{\overrightarrow{D}}_1)=\sigma$, we can determine the charge distributions of modes 1 and 1', as shown in Figs. 3(a)–3(f). Here, mode 1 has like charges while mode 1' has opposite charges in the opposite position of the two graphene nanoribbons. Based on such charge distributions, by the way, the mode 1 and 1' can also be excited by a z-oriented and y-oriented polarized electric dipole source placed in the middle end of the proposed waveguide [9], respectively. When d > 30 nm, a strong E_y distribution is located between the two graphene nanoribbons for both modes, as shown in Figs. 3(b) and 3(c). This indicates that there is a strong interaction between the EM field and the two graphene nanoribbons, which causes additional propagation loss. Thus, the L_prop values of modes 1 and 1' are very small (approximately 3–5 μm), and are only comparable to those of other conventional graphene-based hybrid waveguides [28, 29]. However, with decreasing d, the E_y distribution of mode 1 almost disappears between the two graphene nanoribbons because of the repulsion between like charges, as shown in Fig. 3(e). Thus, the interaction between the EM field and the two graphene nanoribbons decreases rapidly and, hence, the propagation loss is decreased and L_prop is increased significantly. Conversely, the E_y distribution of mode 1' is enhanced owing to the attraction between opposite charges, as shown in Fig. 3(f). Therefore, the interaction between the EM field and the two graphene nanoribbons is increased, which induces a slight increasing in the propagation loss and a decrease in L_prop.

 

Fig. 3 E_y field distributions of modes 1 and 1' for varying d with t = 4 nm. (a, d) modes 1 and 1' E_y field distributions on waveguide cross section for d = 8 nm, respectively. (b, e) mode 1 E_y field distributions on y-z plane for d = 50 and 8nm, respectively. (c, f) mode 1' E_y field distribution on y-z plane for d = 50 and 8nm, respectively.

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Then, we discuss the dependence of the LRSPP-mode characteristics on t. The EM field confinement can be described using the normalized mode area, A_eff /A₀, where A₀ = λ₀²/4. Note that A_eff is the effective mode area, which is defined as $A_{eff}={\displaystyle \int W(r)ds/\max [W(r)]},$ where W(r) is the EM field energy density, given by

The waveguide coupling strength can be set to weak or strong based on d. However, the silica layer plays a different role in different coupling scenarios. For weak coupling (for d > 30 nm, approximately), the waveguide is similar to an ordinary graphene-based hybrid waveguide. The SPP is excited at the two graphene nanoribbons and the electric field decays exponentially at both sides of the two nanoribbons. As the normal boundary conditions of the electric field is ε₁E_1y = ε₂E_2y, a strong electric field is induced in the silica layer due to ε₁ < ε₂, as shown in Fig. 4(e). This is equivalent to transferring the electric field energy from the graphene nanoribbons and silicon layer to the silica layers. Thus, the silica layers play a role of storing energy. With increasing t, more energy can be transferred to these layers. This indicates that the absorption losses of the graphene nanoribbons decrease and the L_prop of the waveguide increases. Simultaneously, the confinement of the EM field increases, whereas (A_eff/A₀) decreases. Obviously, the FOM of the waveguide is improved with increasing t. However, in the strong coupling case (for d < 20 nm, approximately), the majority of the electric field’s energy shifts from the silica layer to the outer space of the waveguide, as shown in Fig. 4(f). In this scenario, the storing-energy capacity of the silica layer is weakened. Note that the waveguide characteristics are primarily determined by the coupling distance. With decreasing t, the coupling distance decreases, which results in a slight improvement in the waveguide performance. On the other hand, whether in a weak-coupling or strong-coupling scenario, the normalized mode area of this waveguide is extremely small (approximately 10⁻⁷ A₀). This is similar to the smallest value reported for another graphene-based hybrid waveguide [29]. The other interesting point to be noted is that the overall performance of the waveguide containing silica layers is clearly superior to that of the waveguide with no silica layers (t = 0 nm), as shown in Figs. 4(a)–4(d).

 

Fig. 4 (a−d) Values of n_r, L_prop, A_eff /A₀, and FOM for mode 1 as functions of d for different t, respectively. (e, f) |E| field distributions of mode 1 located at waveguide y-axis for d = 50 and 8 nm, respectively.

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The L_prop of the graphene-based waveguide can be effectively improved by enhancing the E_f of the graphene. The Fermi level depends on the carrier concentration, which can be controlled via electrical gating or chemical doping. Experimental carrier density values of as high as 10¹⁴ cm⁻² have been reported [41], which is equivalent to E_f = 1.17 eV. Therefore, we set the Fermi level tuning range from 0.4 to 1.0 eV. For comparison, two grahphene-based waveguides, including the LRSPP hybrid waveguide and the normal hybrid waveguide, are investigated with identical parameters of t = 4 nm, d = 6nm, and W = 100 nm. Figure 5(a) shows that (L_prop)_a and (L_prop)_n increase monotonically with E_f. Moreover, with increasing E_f, the increase of (L_prop)_a is more sensitive to E_f than that of (L_prop)_n. Therefore, our proposed waveguide has more potential for improving L_prop, compared with the graphene-based normal hybrid waveguide.

 

Fig. 5 (a) Values of L_prop as function of E_f for LRSPP and NHSPP fundamental modes. (b) L_prop of mode 1 as function of W for different d.

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The L_prop of the proposed waveguide can also be effectively improved by changing W. In this experiment, we set t = 4 nm and E_f = 0.8 eV and vary W from 20 to 180 nm with different d. Figure 5(b) shows that the L_prop of mode 1 is insensitive to W under weak coupling case. However, under strong coupling, the L_prop of mode 1 rapidly increases with W at first, and then gradually tends to a constant value as W > 100 nm. This can be explained as follows. For the LRSPP mode, as mentioned above, the EM fields of the two graphene nanoribbons are mutually exclusive, because of the existing like charges in their opposing positions. Obviously, with increasing W, greater EM-field energy is excluded from the two graphene nanoribbons. Thus, L_prop increases with W. On the other hand, since the majority of the EM field energy is excluded from the waveguide for W > 100 nm, L_prop is then insensitive to W. Hence, we typically select W = 100 nm in our calculations.

Finally, we investigate the dispersion relation of this waveguide. Figure 6(a) and 6(b) show that the n_r values of the LRSPPs and SRSPPs decrease monotonically with increasing λ₀. The higher-order modes have cutoff wavelengths, at which n_r tends toward zero and n_i tends toward infinity. Note that the higher-order modes correspond to smaller cutoff wavelengths and that the SRSPP modes have larger cutoff wavelengths than the LRSPP modes of same mode order. For example, for t = 4 nm, d = 4nm, W = 100 nm, and E_f = 0.8 eV, the cutoff wavelengths of SRSPP modes 2', 3′, and 4' are 16.5, 8.5, and 6.26 μm, respectively. Correspondingly, the cutoff wavelengths of LRSPP modes 2, 3, and 4 are 7, 5.5, and 4.8 μm, respectively. The fundamental modes are cutoff-free. The L_prop values of the different modes have different trend variations with increasing λ₀. For the fundamental modes, the L_prop increases monotonically with λ₀, and mode 1 corresponds to a significantly increasing value. For the higher-order modes, L_prop increases at first and then rapidly drops to zero at the cutoff wavelength. Note that the L_prop values of the higher-order LRSPP modes are significantly smaller than that of mode 1. This can be primarily attributed to the fact that mode 1 and the higher-order LRSPP modes correspond to the graphene SPP edge mode and the graphene SPP waveguide modes [42], respectively, as shown in Figs. 6(d)–6(g). In comparison to the edge mode, the waveguide modes have larger energy confined on the graphene nanoribbon surface, which corresponds to larger propagation losses and smaller L_prop values.

 

Fig. 6 (a, b) n_eff as function of λ₀ for LRSPP and SRSPP modes, respectively. (c) L_prop as function of λ₀ for different modes. (d, e) E_y field distribution of modes 2 and 3 on waveguide cross section, respectively. (f, g) |E| field distribution of modes 2 and 3 on waveguide surface, respectively.

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4. LRSPP hybrid mode in proposed waveguide with substrate

For practical applications, we place the graphene-based LRSPP hybrid waveguide on a buffer layer of thickness h on top of a substrate, as shown in Fig. 7(a). In this case, the effective index matching is destroyed because of the asymmetric waveguide profile. Then, the coupling strength of the GSPPMs inevitably decreases. The waveguide EM-field distribution becomes asymmetric, as shown in Fig. 7(b) and its insets. From Fig. 7(c), it is apparent that the L_prop of the asymmetric waveguide is smaller than that of the symmetric waveguide discussed in previous research, which results in a decrease in performance. Nonetheless, the asymmetric waveguide has similar LRSPP mode properties to the symmetric waveguide. Moreover, the asymmetric waveguide has a significantly increased L_prop and a comparable ultra-small mode area compared to other conventional graphene-based hybrid waveguides [28,29].

 

Fig. 7 (a) Proposed waveguide placed on a buffer layer on substrate. (b) |E| distribution of mode 1 at asymmetric waveguide y-axis, and its two insets show |E| and E_y distributions on asymmetric waveguide cross section, respectively. (c) L_prop and A_eff /A₀ of mode 1 as functions of d for asymmetric and symmetric waveguides.

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5. Conclusion

In summary, we have proposed and investigated a graphene-based LRSPP hybrid waveguide, which is composed of two identical outer graphene nanoribbons and two identical inner silica layers symmetrically placed on both sides of a silicon layer. The coupled mode perturbation theory shows that the LRSPP and SRSPP modes originate from the coupling of the same modes of the two graphene nanoribbons, which agrees well with the numerical simulation results. Compared with the normal graphene-based hybrid waveguide design, the proposed waveguide with the LRSPP fundamental mode has a distinct advantage that it can simultaneously achieve an ultra-long propagation length (~10 μm) and an ultra-small mode area (~10⁻⁷A₀) through variation of the silica- and silicon-layer thicknesses, the width of the waveguide, and the Fermi energy of the graphene. This waveguide can be used to construct various functional devices for guiding mid-infrared SPP waves in future photonic integrated circuits.