Design of a broadband single mode hybrid plasmonic waveguide incorporating silicon nanowire

K. B. M. Rakib Hasan; K. B. M. Rakib Hasan; Md. Asiful Islam; M. Shah Alam

doi:10.1364/OME.405037

1. Introduction

Surface plasmon polariton (SPP) wave, an electromagnetic wave evanescently confined to the metal-dielectric interface, has presented considerably high promise in recent times for its subwavelength light-guiding capability and unprecedented optical bandwidths [1–5]. Plasmonic waveguides are thought as replacements of the traditional metallic interconnects in VLSI circuits for realizing compact photonic integrated circuits [4–6]. Although the SPP waveguides are reported for a deep subwavelength device size, they suffer from high optical loss [1,6]. This can be attributed to the high optical confinement in the dissipative metallic region [1]. Hence, to tackle this major drawback of plasmonic waveguides, a new hybrid waveguide currently has grown significant research interest among the plasmonic researchers. Here, a hybrid electromagnetic mode is evolved from the mode hybridization between SPP mode and dielectric photonic mode, which offers a high mode confinement in the low index dielectric region [6–8]. This low index region is sandwiched between a metallic region and high index dielectric region. These waveguides are commonly known as hybrid plasmonic waveguides, which offer both long range light propagation and subwavelength device footprint simultaneously [6–12].

The integration of graphene into the design of plasmonic waveguides, is being also noticed in many literatures to obtain various functionalities of photonic integrated circuits [13–20]. The main reason behind the research trend can be attributed to the exotic electronic and optical properties of graphene [21,22]. The intriguing electronic properties are due to a slight overlap between the conduction band and valence band at the Dirac point [21,22]. Furthermore, the experimentally reported controllability of graphene’s interband transition through electrical gating, leads to a unique tunable optical absorption [21–23]. This has been exploited in various applications like optical switching, modulation, and long-range propagation of SPP waves [2,13–23]. In graphene-based modulators or switches, the optical absorption of graphene can be electrically controlled to act it as both metallic (high optical absorption) and dielectric (low optical absorption) at a given operating wavelength [13–20,23], which enables the intensity modulation of optical signal with a highspeed electrical bitstream. Moreover, graphene possesses a high theoretical modulation bandwidth (∼500 GHz) making it a promising candidate for highspeed modulation and switching applications [22]. Also, graphene supports surface plasmon modes, when it acts as a metal. The fact that the SPP mode is highly confined in graphene layers, can be exploited in designing promising optical interconnects with deep subwavelength feature size as an alternative of conventional metallic SPP guides [2,10,14].

Among various designs of plasmonic waveguides, the metallic nanowire based guiding geometries have been studied to a great extent for rendering different functionalities [3,5,6,12,19,23–26]. Double metallic nanowire system with a dielectric cladding region, has been studied in detail to exploit the multimode operability in designing plasmonic mode converters [25]. The study also shows that the single metal nanowires with moderate radii support several higher order modes which couple to give rise to a number of higher order modes in a double nanowire system [25]. Apart from that, metallic and dielectric nanowires have been employed in designing different hybrid plasmonic waveguides to obtain deep subwavelength mode confinement and small optical loss simultaneously [3,6,12,26]. Moreover, the graphene coated dielectric nanowires are reported to support SPP modes as in the single metallic nanowires [27,28]. However, it is required to design single mode plasmonic nanowire waveguides to get rid of the impairments related to the intermodal dispersion that occurs in the multimode waveguide. Therefore, the single mode SPP waveguides could be realized for optical data transmission in nanoscale as an analogue to single mode fibers in optical fiber communication system. One way to make the single metallic nanowires single mode plasmon waveguides, is to reduce the nanowire radii. But the extreme scaling down of the nanowire radius possibly imposes a challenge on the traditional fabrication processes [29]. Furthermore, among all the SPP modes, only the quadrupole SPP mode is cut off in nanowires of moderate radii (∼50 nm) [24,25]. So, the fundamental monopole and two degenerate dipole modes with orthogonal polarizations still exist in single metallic nanowire system leading to chiral surface plasmon polaritons [30]. The excitation of chiral SPPs are reported in [30], where it is generated by linearly polarized light being incident at one end of the nanowire. So, there is still need for research to obtain single mode SPP guide from the chiral SPPs. Previously, the possibility of achieving monopole SPP waveguide has been demonstrated using graphene-coated hybrid plasmon waveguides [27,28]. Nevertheless, the dipole modes possess antisymmetric electric field component, which makes them long-range SPP modes of the system [31]. The propagation length of the dipole modes increases for decreasing nanowire radius, while the propagation length of the monopole mode shows the opposite trend [31]. Thus, the dipole modes are more promising in long-range single mode optical information transmission than the monopole SPP mode. Apart from that in the design of a plasmonic network with metallic nanowires enabling mode division multiplexing (MDM), the single mode operability of metallic nanowires needs to be realized for all the SPP mode of our system. This is required for (de)multiplexing of optical information in nanoscale, where each mode acts as a carrier of a specific optical information just like the silicon photonic MDM system proposed in [32]. Hence, the design of single mode SPP waveguide is equally essential for both the monopole and dipole modes.

In this paper, we have proposed a graphene hybrid plasmonic waveguide for the possible single mode operation of the y-polarized dipole SPP mode. The graphene layers in our design extinguish the undesired monopole and x-polarized dipole SPP mode. In addition to that the figure of merit of the desired y-polarized dipole mode, has been enhanced forming a hybrid plasmonic waveguide with gold and silicon nanowires. Here, the figure of merit has been defined as the ratio of mode propagation length and effective diameter of mode cross-section [28]. The eigen mode analysis also reveals that the performance of our proposed waveguide does not deteriorate for a broad wavelength range containing the telecommunication wavelengths. Thus, the proposed graphene hybrid plasmonic waveguide shows a high promise in nanoscale optical information transmission.

2. Design of a waveguide and analysis method

The schematic of the proposed graphene hybrid plasmonic waveguide (GHPW) for possible broadband single mode operation is shown in Fig. 1. Two different perspectives of the waveguide model have been presented in Fig. 1(a) and Fig. 1(b), to clearly illustrate its geometry. In this design, the gold (Au) nanowire supports the surface plasmon modes propagating along the z-axis, and thus the nanowire is considered parallel to the z-axis. As the Au nanowire is designed to support only the y-polarized dipole mode, two silicon (Si) nanowires are symmetrically located around the central Au nanowire along the same y-z plane to form a hybrid plasmon waveguide. The Si nanowires are also parallel to the z-axis as shown in Fig. 1. All these nanowires have identical cross-section with radius, r = 50 nm. The inter nanowire gap width along the y-axis on either side of the Au nanowire has been denoted as d₂ in the diagram. The graphene-hexagonal Boron Nitride-graphene (graphene-hBN-graphene) layers are considered to be composed of a 20 nm thick hBN layer to be sandwiched between two graphene monolayers. The reason for using hexagonal boron nitride is that the highest ever mobility of graphene has been reported for graphene on hBN substrate with strongly suppressed charge inhomogeneities [33]. Also, hexagonal boron nitride is a layered material with the same planar atomic structure as graphene and an only 1.8% longer lattice constant than graphene [33]. These layers are uniformly located around the Au nanowire perpendicular to the x-z plane. The Au electrodes are assumed here for applying a constant electrical bias voltage at a fixed operating wavelength to maximize the optical absorption of the graphene layers. Each graphene-hBN-graphene layer having a width of W = 300 nm, is d₁ distance away from the Au nanowire as depicted in Fig. 1. All these nanostructures are embedded in silica (SiO₂) cladding region. In this design, we have assumed that a multimode plasmon excitation enters into the GHPW through one end (input port) of the Au nanowire, and after passing through the L_d ( = 10 µm) long waveguide region at the other end (output port) all other SPP modes are expected to extinguish except for the only y-polarized dipole SPP mode. Therefore, the separation between the input and output ports is the device length (L_d).

Fig. 1. Schematic of the proposed waveguide structure with the isometric projection along (a) the z-axis, and (b) y-axis.

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In order to fabricate the proposed nanostructure, the hBN layers could firstly be grown on silica using plasma-assisted chemical vapor deposition [34]. The graphene monolayers could also be grown on each side of an hBN layer applying the same technique in addition to exfoliating the other side onto a Si/SiO₂ substrate layer by mechanical cleaving [34]. Then, a SiO₂ layer of nanoscale thickness can be deposited on top of the graphene-hBN-graphene layer using the deposition method used in [35]. After that the Au and Si nanowires can be placed on the deposited SiO₂ layer. The inter nanowire gaps between the central Au wire and the adjacent Si wires could be well controlled using available self-assembly chemical fabrication methods [36]. Firstly, the Au wire can be coated with a homogeneous SiO₂ layer and then attaching/binding the adjacent Si nanowires to the coated Au nanowire, which can result in well-controlled gap widths [36]. To obtain more precise control on the inter nanowire gap width and alignments, optical tweezers could be used [36]. Next, SiO₂ layer can be deposited again to form the cladding region of the waveguide by immersing all the nanowires. Finally, another graphene-hBN-graphene layer can also be grown on top of SiO₂ like the previous one [34]. The probable small airgaps resulted from the mismatch of cross-sections between the circular and planar geometries, can possibly be addressed by air vaporization using SiO₂ annealing [37]. The Au nanowire in our design can possibly be fabricated by a three-step seed-mediated method followed by a purification of the high aspect ratio Au nanorods as in [38,39], while the Si nanowires could be synthesized using the vapor-liquid-solid (VLS) growth as in [36].

The proposed GHPW has been designed in COMSOL Multiphysics and the E-field based full-vectorial finite element method (VFEM) is used here to obtain the modal parameters [5,23]. To formulate the cross-section of the waveguide, geometry controlled triangular mesh with a minimum mesh dimension of 0.022 nm is used. A 250 nm thick perfectly matched layer (PML) has been considered at the outer boundary of SiO₂ cladding region. The outer edge of the PML is perfect electric conductor (PEC). As our design is a subwavelength structure, the radius of the SiO₂ cladding region has been regarded to be six times of the operating wavelength in this simulation. The refractive index of the plasmonic material Au has been derived from [40], where the Drude Model, interband transition of metal, and size dependent damping frequency have been incorporated. The refractive indices of Si and SiO₂ have been obtained from the Palik data [41]. The hBN layers have been modelled as anisotropic material and the optical properties are calculated from the analytical equations used in [22,23,42].

The dielectric function of each graphene monolayer (ε_g) can be derived from its surface conductivity (σ_g) using the following equation [2,13,19,23];

(1)$${\varepsilon _g}(\omega ) = 1 + \frac{{i{\sigma _g}(\omega )}}{{{\varepsilon _0}\omega {d_g}}},$$

where, ω is the angular frequency, ε₀ is the free space permittivity, and d_g is the thickness of the graphene monolayers. In this simulation, d_g = 1 nm has been assumed for the ease of simulation as in [13,22,23]. The surface conductivity of graphene can be determined employing the Kubo formalisms, which considers the contributions of both the interband and intraband transitions [13,22,23];

(2)$$\begin{array}{l} {\sigma _g}(\omega ) = \frac{{2i{e^2}{k_B}T}}{{\pi {\hbar ^2}(\omega + i{\tau ^{ - 1}})}}\ln \left[ {2\cosh \left( {\frac{{{\mu_c}}}{{2{k_B}T}}} \right)} \right] + \\ \frac{{{e^2}}}{{4\hbar }}\left\{ {\frac{1}{2} + \frac{1}{\pi }{{\tan }^{ - 1}}\left( {\frac{{\hbar \omega - 2{\mu_c}}}{{{k_B}T}}} \right) - \frac{i}{{2\pi }}\ln \left[ {\frac{{{{(\hbar \omega + 2{\mu_c})}^2}}}{{{{(\hbar \omega - 2{\mu_c})}^2} + {{(2{k_B}T)}^2}}}} \right]} \right\}, \end{array}$$

Here, e is the charge of electron, ħ is the reduced Planck constant, k_B is the Boltzmann constant, T is the temperature, and τ is the inter-band relaxation time. The inter-band relaxation time has been regarded to be 20 fs as reported in [43]. The chemical potential μ_c is a function of the gate bias voltages and can be obtained from [22];

(3)$${\mu _c} = \hbar {v_F}\sqrt {\pi {n_0}} = \hbar {v_F}\sqrt {\frac{{\pi {\varepsilon _0}{\varepsilon _r}|{{V_g} - {V_{Dirac}}} |}}{{{d_{hBN}}e}}} .$$

where, the Fermi velocity of electron has been denoted as v_F (≈ 1.1×10⁶ ms⁻¹) and the carrier concentration in the graphene sheets is n₀ [22,23]. For calculating n₀, each graphene-hBN-graphene layer has been regarded as a parallel-plate capacitor formed by two graphene layers with a d_hBN ( = 20 nm) thick intermediate hBN layer [22,23]. Thus, a relation between μ_c and gate bias voltage (V_g) is established in Eq. (3). Also, V_Dirac is the notation of the offset voltage due to the natural doping [22,23]. The value of V_Dirac is typically around 0.8 V for the structures containing an hBN layer sandwiched between two monoatomic graphene layers [44]. However, in this work, we have obtained the eigenmode solutions only considering the variation of chemical potential. From Eq. (1), Eq. (2), and Eq. (3), we find the isotropic refractive index of graphene to be the function of both the operating wavelength and chemical potential. The dispersion of graphene's optical function for the variation of chemical potential at room temperature (T = 300 K) with the operating wavelength to be the telecommunication wavelength (λ = 1.55 µm) has been shown in Fig. 2. Here, we see that the real part of dielectric function is positive for µ_c < 0.53 eV, which means the graphene monolayers exhibit dielectric properties. At µ_c = 0.53 eV, the optical property of graphene experiences a transition from dielectric to metallic behavior, as the dielectric function becomes negative [22,23]. Again, the imaginary part of dielectric function monotonically decreases for increasing µ_c. So, the optical absorption of graphene maximizes at µ_c = 0.53 eV for telecom wavelength. Thus, the corresponding gate bias (V_g) should be applied to the graphene layers at λ = 1.55 µm, in order to achieve the highest possible mode extinction for the undesired plasmon modes supported by the Au nanowire. Also, it can be intuitively seen that the chemical potential (or V_g) at which graphene’s optical absorption reaches its peak, changes considerably with the variation of the operating wavelength, which will be shown later.

Fig. 2. Dielectric function of graphene for different chemical potentials at the telecommunication wavelength for T = 300 K

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In this analysis, firstly we have studied the modal properties of different SPP modes supported by a single Au nanowire (r = 50 nm) with SiO₂ cladding for a broadband of operating wavelength in the infrared regime. Then, we have considered the Si nanowires and graphene-hBN-graphene layers around the Au nanowire to form the proposed waveguide structure and explored the modal solutions. Here, our design objective is not only to extinguish the undesired SPP modes but also to increase the figure of merit of the desired SPP mode. Hence, we have compared the calculated modal parameters of the first one with those of the latter one for varying cross-sectional geometry and operating wavelength. A promising performance has been demonstrated by our design in this regard. Finally, the effect of graphene’s optical anisotropy on the waveguide performance has been discussed very briefly.

3. Results and discussions

In this section, at first, we will present the eigen mode analysis of a single Au nanowire with SiO₂ cladding. Here, we will describe the modal characteristics of the undesired SPP modes in contrast to the desired dipole SPP mode in our system. Then we will show how our proposed design can suppress the multimode operation and enhance the figure of merit of the only desired y-polarized dipole mode.

3.1. SPP modes of a single Au nanowire in SiO₂ cladding

From the previous literature reviews on single metallic nanowire based plasmonic waveguides, metallic nanowires with moderate radii (∼100 nm) support monopole, dipole and quadrupole plasmon modes [25]. However, for smaller radius the quadrupole mode is cutoff for infrared wavelength, and the metal nanowire supports only the monopole and dipole SPP modes [25]. The dipole modes evolve as two degenerate modes having orthogonal polarizations (x and y polarized SPP modes) [24,25]. The |E| field distributions and the dominant E_z field profiles of the SPP modes supported by a single Au nanowire embedded in SiO₂ cladding region at λ = 1.55 µm are shown in Fig. 3. Here, the radius r of Au nanowire has been considered to be 50 nm as stated earlier. The monopole and dipole modes have been denoted as m = 0 mode and m = 1 mode, respectively as in [25]. As the surface plasmon modes are the longitudinal collective oscillations of the surface electron, usually they are quasi-TM in nature and among the all three components of electric field only the z component is dominant [1,25]. This is why, only E_z profile has been shown in Fig. 3. The m = 0 mode is the fundamental mode and its surface charge is uniformly distributed over the Au nanowire surface, which is demonstrated by Fig. 3(a) and Fig. 3(d). For m = 1 mode, the electromagnetic interactions along both the x and y axes are identical. This gives rise to two degenerate x-polarized and y-polarized dipole modes with same effective refractive index as shown in Fig. 3(b) and Fig. 3(c). Figure 3(e) and Fig. 3(f) shows the evident of the stated dipole nature of the m = 1 mode, as each mode has two lobes containing surface charges with opposite polarities. However, these modes show much weaker mode confinement at the telecom wavelength than the fundamental m = 0 mode, which can be qualitatively attributed to a low figure of merit of the desired mode to be shown later.

Fig. 3. |E| field distributions of (a) monopole (m = 0) mode, (b) x-polarized dipole (m = 1) mode, and (c) y-polarized dipole (m = 1) SPP mode; E_z field profile of (d) monopole (m = 0) mode, (e) x-polarized dipole (m = 1) mode, and (f) y-polarized dipole (m = 1) SPP mode on the Au nanowire with SiO₂ cladding at telecom wavelength, where the Au nanowire radius r = 50 nm.

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The quantitative picture of the modal characteristics of this waveguide becomes clear by calculating effective mode index (n_eff), propagation length (L_p), and effective mode area (A_eff). The variation of these parameters is calculated using following equations as in [6,25];

(4)$${L_p} = \frac{1}{{2{K_{eff}}{k_0}}},$$

(5)$${A_{eff}} = \frac{{\int_{ - \infty }^\infty {W(x,y)dxdy} }}{{\max [W(x,y)]}},$$

where, K_eff denotes the imaginary part of effective refractive index, k₀ ( = 2π/λ) is the free space wavenumber, and the maximum value over the waveguide cross-section is indicated as max[.]. W(x,y) is the energy density and it is calculated using the magnitudes of electric and magnetic fields as [25],

(6)$$W(x,y) = \frac{1}{2}{\textrm{Re}} \left\{ {\frac{{d[\omega {\varepsilon_0}{\varepsilon_r}(x,y)]}}{{d\omega }}} \right\}{|{E(x,y)} |^2} + \frac{1}{2}{\mu _0}{|{H(x,y)} |^2}.$$

From Fig. 4, the effective index of m = 0 mode is higher than that of m = 1 mode for any operating wavelength, as the m = 0 mode is the fundamental SPP mode in our system. The x and y polarized m = 1 modes are degenerate modes and hence, they have the same effective index as shown in Fig. 4. Also, the effective indices show very little variation throughout the given range of operating wavelength. This can be correlated to the dispersions of other modal parameters. The propagation length and effective area also shows a very little variation too as shown in Fig. 4. As stated earlier, the fundamental m = 0 mode is highly confined to Au-SiO₂ interface and consequently having subwavelength modal area at the telecom wavelength (∼0.038A₀). However, this mode is not appropriate for the propagation of optical information for the reported high modal attenuation. As an evident, we find its propagation length to be only 6 µm at the telecom wavelength. The m = 1 mode seems to be little more promising than the fundamental mode for its slightly longer propagation length (∼13 µm) at the telecom wavelength. Nevertheless, the large effective mode size of this mode prevents the appropriateness of this optical waveguide in compact photonic integrated circuit design. Moreover, the propagation length monotonically decreases with increasing wavelength, while the effective mode size gets larger consequently resulting in low figure of merit. However, the demerit has been tackled efficiently in some literature by utilizing this geometry in sensing applications which require high optical confinement in the dielectric cladding region [24]. As our design objective is to achieve high compatibility of the y-polarized m = 1 mode in subwavelength light guiding applications with negligible intermodal dispersion, we will show now how the proposed design can serve the purpose.

Fig. 4. Real part of effective index (Re(n_eff)), effective mode size (A_eff), and propagation length (L_p) of different SPP modes supported by single Au nanowire in SiO₂ cladding for different operating wavelengths; where, A₀ is the diffraction limited effective mode area (= λ²/4).

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3.2. Modal characteristics of the proposed GHPW for variable gap widths

In this section, we have varied the gap widths d₁ and d₂ to examine effects on the modal properties of the GHPW at telecom wavelength. The electric field distributions of different SPP modes supported by the GHPW at telecom wavelength have been presented in Fig. 5. Here, the locations of the graphene-hBN-graphene layers coincide with the location of optical intensity of the m = 0 mode, and x-polarized m = 1 mode as shown in Fig. 5(a) and Fig. 5(b). Therefore, the properly biased (µ_c = 0.53 eV) graphene-hBN-graphene layers should significantly reduce the propagation distances of these modes due to the peak optical absorption of graphene. In that case, the optical intensity is highly confined in the graphene monolayers. On the other hand, the desired y-polarized m = 1 mode has the minimum interaction with the graphene layers thus having maximum propagation length as shown in Fig. 5(c). Moreover, this mode renders subwavelength modal area with large propagation distance, as it is confined in the inter nanowire gap regions between Au and Si nanowires. The Si nanowires with the central Au nanowire forms a hybrid plasmonic waveguide, where high mode confinement is obtained in the low index SiO₂ region due to the optical capacitance effect [6,23]. The polarization charge at the Si-SiO₂ interface is optically coupled with the electron plasma oscillation and thus the improve propagation characteristics of y-polarized dipole mode are achieved [6]. So, if a multimode excitation enters the input port, all other SPP modes, except the y-polarized m = 1 mode, are expected to drop their optical intensities significantly while passing through the 10 µm long waveguide regions. Hence, our proposed waveguide is expected to extinguish the undesired modes at the output port. However, the gap width has remarkable effects on its performance.

Fig. 5. |E| field distribution of the (a) m = 0 mode, (b) x-polarized m = 1 mode, and (c) y-polarized m = 1 mode supported by the proposed GHPW at the telecommunication wavelength for d₁ = d₂ = 10 nm.

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The variations of the real part of effective index and propagation length have been shown in Fig. 6 for different d₁ and d₂ at the telecommunication wavelength. While d₁ (or d₂) is varied, d₂ (or d₁) is assumed to be fixed at 10 nm. Both the effective indices and propagation lengths of all the modes remain almost unchanged for large variation of d₁ except the fundamental m = 0 mode as depicted in Fig. 6(a) and Fig. 6(b). Although the fundamental m = 0 mode encounter an increase in propagation length (from 1.6 µm to 2 µm) for increasing d₁, it remains significantly smaller than that of the desires y-polarized dipole SPP mode. The propagation length of the y-polarized m = 1 mode is about 23 µm, while the other dipole mode has propagation length of about 9 µm for a wide range of d₁. Therefore, the gap width between the Au nanowire and each graphene-hBN-graphene layer has very little effect on the waveguide performance. Thus, the graphene-hBN-graphene layers need not be placed symmetrically around the central Au nanowire at its close vicinity. This is how, the design offers a possible flexibility in the fabrication of the graphene-hBN-graphene layers. However, d₁ should be kept as small as possible to achieve a deep subwavelength device footprint of the GHPW. Unlike d₁, d₂ has a significant impact on the performance of the proposed GHPW. For increasing d₂, the optical capacitance effect gets weaker. This affects the mode confinement in the gaps between the Au and Si nanowires consequently reducing the effective refractive index and propagation length of the y-polarized m = 1 mode as shown in Fig. 6(c) and Fig. 6(d). On the other hand, the m = 0 mode shows an increase in propagation length in spite of its decreasing real part of effective index for larger d₂. However, the x-polarized m = 1 mode has negligible optical intensity in the Si-Au nanowire gap regions. So, the alteration of d₂ has almost no effect on the effective refractive index and propagation length of this mode. As the propagation length of the desired SPP mode decreases monotonically for increasing d₂, the Si nanowires should be located at closest possible distance from the central Au nanowire to achieve a better performance. It is surely possible to consider d₁ and d₂ less than 10 nm in our proposed design, which would further improve the waveguiding performance. However, the smaller the feature size in a nanostructure, the more the fabrication complexities. Thus, the results have been shown up to d₁ = d₂ = 10 nm.

Fig. 6. (a) Real part of effective index (Re(n_eff)), (b) propagation length (L_p) of different SPP modes in the GHPW for variable d₁ and d₂ = 10 nm; (c) real part of effective index (Re(n_eff)), (d) propagation length (L_p) of different SPP modes in the GHPW for variable d₂ and d₁ = 10 nm at telecom wavelength, and (e) schematic of waveguide cross-section.

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The variation of effective mode area of the SPP modes for different d₁ and d₂ have been shown in Fig. 7(a) and Fig. 7(b). The deep subwavelength effective mode areas of these modes remain almost constant for larger gap widths. In order to get a more quantitative picture of single mode operation of the proposed GHPW, the extinction ratios (ER) of the undesired SPP modes to the desired mode are considered here. Mathematically, ER is the ratio of the output optical intensity of the desired mode to that of an undesired mode in logarithmic scale. As our desired mode is y-polarized m = 1 mode and the other two modes are the undesired modes of system, the ERs are defined as follows;

(7)$$E{R_1} = 10\log \left( {\frac{{{P_{m = 1,y}}}}{{{P_{m = 0}}}}} \right) = \frac{{109.15{L_d}({k_{m = 0}} - {k_{m = 1,y}})}}{\lambda },$$

and

(8)$$E{R_2} = 10\log \left( {\frac{{{P_{m = 1,y}}}}{{{P_{m = 1,x}}}}} \right) = \frac{{109.15{L_d}({k_{m = 1,x}} - {k_{m = 1,y}})}}{\lambda },$$

where, P_m=0 is the optical intensity received at the output port of m = 0 mode, P_m=1,x is the output optical intensity of x-polarized m = 1 mode, P_m=1,y is the output optical intensity of y-polarized m = 1 mode, k_m=0 is the imaginary part of effective index of m = 0 mode, k_m=1,x is the imaginary effective index of the x-polarized m = 1 mode, and k_m=1,y is the imaginary effective index of the desired y-polarized m = 1 mode. Clearly, ER₁ is the extinction ratio of m = 0 mode to the desired SPP mode of the system and ER₂ is the extinction ratio of the x-polarized m = 1 mode. So, the larger the ERs, the better the performance. If I₀ is the input optical intensity of each SPP mode in our system, the P_m=0, P_m=1,x, and P_m=1,y can be written as follows [19];

(9)$${P_{m = 0}} = {I_0}{e^{ - \frac{{4\pi {k_{m = 0}}{L_d}}}{\lambda }}},$$

(10)$${P_{m = 1,x}} = {I_0}{e^{ - \frac{{4\pi {k_{m = 1,x}}{L_d}}}{\lambda }}},$$

and

(11)$${P_{m = 1,y}} = {I_0}{e^{ - \frac{{4\pi {k_{m = 1,y}}{L_d}}}{\lambda }}}.$$

The ER₁ and ER₂ values for the variation of d₁ and d₂ have been shown in Fig. 7(c) and Fig. 7(d). Although ER₂ slightly increases for increasing d₁, ER₁ noticeably reduces as demonstrated in Fig. 7(c). On the other hand, both ER₁ and ER₂ monotonically reduces for increasing d₂, which is shown in Fig. 7(d). This can be attributed to the reduction of the strength of optical capacitance effect as stated earlier. Therefore, our proposed GHPW shows its optimum performance at single mode operation with ER₁ = 49 dB and ER₂ = 6 dB for d₁ = d₂ = 10 nm at the telecommunication wavelength.

Fig. 7. Effective mode area (A_eff) of the SPP modes in GHPW for (a) variable d₁ and d₂ = 10 nm, (b) variable d₂ and d₁ = 10 nm; extinction ratios (ERs) of the undesired SPP modes for (c) variable d₁ and d₂ = 10 nm, (d) variable d₂ and d₁ = 10 nm at telecom wavelength. Here, A₀ (=λ²/4) is the diffraction limited effective area and L_d = 10 µm.

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3.3. Multiwavelength operability of the proposed GHPW

As stated earlier, the dielectric function of graphene is a strong function of both the operating wavelength and chemical potential. The chemical potential dependence of the real part of graphene’s dielectric function for different operating wavelengths, has been shown in Fig. 8(a). Here, the chemical potential at which the graphene’s optical absorption maximizes, shifts towards higher chemical potential for decreasing the operating wavelength. This is due to the fact that graphene has a higher shift point [22]. So, the required chemical potential (or electrical bias) for achieving highest possible mode extinction of the undesired modes, decreases monotonically for increasing operating wavelength as shown in Fig. 8(b). Now we will show how ER₁ and ER₂ changes for altering the operating wavelength, where the corresponding µ_c values for different operating wavelengths have been taken from Fig. 8(b).

Fig. 8. (a) Real part of graphene’s dielectric function versus chemical potential for different operating wavelengths, (b) The chemical potential and V_g –V_Dirac at which graphene’s optical absorption maximizes for different operating wavelengths.

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The wavelength dependence of ER₁ and ER₂ for different gap widths has been shown in Fig. 9. While one gap width is varied, the other is fixed at 10 nm. From Figs. 9(a) and 9(b), the trend of wavelength dependence of ER₁ is the same regardless of gap widths. At smaller wavelengths, a higher ER₁ is obtained and it decreases monotonically for decreasing wavelength. Also, d₂ has almost no effect of the value of ER₁ throughout the given wavelength range, while d₁ affects the value of ER₁ at smaller wavelengths. For d₁ = 10 nm, the ER₁ is about 100 dB at λ = 1.2 µm as shown in Fig. 9(a). Conversely, the trend of wavelength dependence of ER₂ changes significantly for alteration of d₂ as shown in Fig. 9(d). The peak ER₂ shifts towards smaller wavelength for increasing d₂, while d₁ = 10 nm. However, when d₂ is fixed at 10 nm, the variation of d₁ does not change the location of peak ER₂, rather the maximum ER₂ increases with decreasing d₁. Since the device performance is optimum at d₁ = d₂ = 10 nm for the telecommunication wavelength, the maximum ER₂ (∼10 dB) is obtained at λ = 1.35 µm. For larger wavelengths, a lower ER₂ is obtained as shown in Figs. 9(c) and 9(d), which implies that the optical intensities of the x and y-polarized dipole modes are comparable in order of magnitude according to Eq. (8). In that case, the undesired polarization of the dipole SPP mode will be mixed with the desired polarization state. It will possibly result in circularly polarized SPP mode that will propagate helically along the nanowire surface [31]. This is expected to trouble the use of our proposed waveguide for single mode operation with only the one polarization state of the dipole SPP mode. Although the ER₁ significantly high for 1.2 µm < λ < 1.6 µm, the ER₂ is not high enough to that extent. In order to improve ER₂, we need to consider smaller d₁ than 10 nm. However, it would increase the complexity of fabrication of the proposed device. One way to circumvent the problem is to use a multilayer stacking of graphene and hBN layers and hence enhancing the optical absorption of the proposed device as in [13]. This would possibly increase the value of ER₂ in the given range of operating wavelength. Therefore, our proposed design shows promising mode ER₁ throughout the telecommunication window for d₁ = d₂ = 10 nm and has the potential to achieve promising ER₂ by adding a graphene multilayer stack. The simulation results for different number of graphene monolayers in the graphene-hBN multilayer stack located at each inter nanowire gap are shown in Table 1, where λ = 1.55 µm and d₁ = d₂ = 10 nm are considered. Here, both ERs substantially increase for a higher number of graphene monolayers for an enhanced optical absorption.

Fig. 9. Wavelength dependence of the extinction ratio of monopole mode (ER₁) for (a) varying d₁ and fixed d₂ ( = 10 nm), (b) varying d₂ and fixed d₁ ( = 10 nm); wavelength dependence of the extinction ratio of x-polarized dipole mode (ER₂) for (c) varying d₁ and fixed d₂ ( = 10 nm), (d) varying d₂ and fixed d₁ ( = 10 nm). Here, L_d= 10 µm is considered.

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Table 1. Extinction ratios of undesired SPP modes for graphene multilayer stack, where λ = 1.55 µm and d₁ = d₂ = 10 nm

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In the previous sections, we have analyzed the performance of the proposed waveguide in extinguishing the undesired modes. Now, we will show how much this design can enhance the figure of merit (FOM) of the desired y-polarized dipole SPP mode. The FOM has been calculated as the ratio of mode propagation length and effective diameter of mode cross-section, which can be expressed as follows [28];

(12)$$FOM = \frac{{{L_p}}}{{2\sqrt {{{{A_{eff}}} / \pi }} }}.$$

The FOMs of the y-polarized dipole mode in single Au nanowire and the proposed GHPW are calculated using Eq. (12) and shown in Fig. 10. Clearly, the desired y-polarized m = 1 mode in the proposed GHPW demonstrates a way larger FOM than that of the same mode supported by the single Au nanowire with SiO₂ cladding throughout a long wavelength range. Therefore, a high figure of merit of the desired SPP mode for a broadband of operating wavelength, which covers the telecommunication window. Most of the proposed graphene-based nanowire waveguides are reported to operate in the THz regime. So, in order to present a fair comparison, normalized figure of merit (FOM_norm) has been calculated for some of the previously proposed works and presented in Table 2. The FOM_norm of our structure is still promising compared to the other graphene-based nanowire waveguides.

Fig. 10. Wavelength dependence of the FOMs of the y-polarized dipole SPP (m = 1) mode in single Au nanowire and the proposed GHPW structures.

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Table 2. Normalized figure of merit for different nanowire-based graphene plasmonic waveguides

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3.4. Effects of graphene’s anisotropy on waveguiding performance

Previously, we have considered isotropic optical properties of graphene to obtain the mode analysis of the proposed structure. However, graphene has been reported to demonstrate optical anisotropy; where only its in-plane permittivity can be derived from Kubo formalisms using Eq. (1) and Eq. (2), while the out-of-plane permittivity remains almost constant (∼2.5) [23,48–50]. This should affect the waveguiding performance. Therefore, we have considered chemical potential dependence of the extinction ratios of undesired SPP modes (ER₁ and ER₂) for three different operating wavelengths considering the mentioned anisotropic optical properties of graphene and shown in Fig. 11(a) and Fig. 11(b). Here, a higher ER₁ is obtained for smaller chemical potential and the maxima are found at µ_c = 0 eV as shown in Fig. 11(a), regardless of operating wavelength. On the other hand, ER₂ reach its maximum value at a higher chemical potential for each of the operating wavelengths, where the corresponding µ_c at which ER₂ maximizes shifts towards a larger value for increasing λ as shown in Fig. 11(b). However, the peak ER₂ is only slightly higher (less than 1 dB) than its value at µ_c = 0 eV around the λ of our interest. So, µ_c = 0 eV can be considered as the operating point regardless of the operating wavelength and undesired SPP mode of interest. This has been considered for obtaining the results of Fig. 11(c). Here, the ERs are still promising for anisotropic graphene layers, while there is essentially no need to apply any electrical bias to the graphene layers for a specific operating wavelength unlike the isotropic graphene layers given that V_Dirac is sufficiently small. Thus, the ER₁ and ER₂ have been recalculated considering µ_c = 0 eV for a wide range of infrared wavelengths around the telecom wavelength and shown in Fig. 11(c).

Fig. 11. Chemical potential (µ_c) dependences of (a) extinction ratio of m = 0 mode (ER₁), (b) extinction ratio of x-polarized m = 1 mode (ER₂), for three different operating wavelengths around the telecom wavelength, and (c) wavelength dependence of ER₁ and ER₂, while µ_c = 0 eV has been assumed. Here, graphene is considered an anisotropic material and d₁ = d₂ = 10 nm for all three cases.

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Here, the device shows a promising performance in extinguishing the undesired SPP modes throughout the telecommunication window. Furthermore, the figure of merit of the desired dipole SPP mode should remain unaffected by graphene’s optical anisotropy, as the mode has a negligible optical field confinement in the graphene-hBN-graphene layers.

4. Conclusion

We have demonstrated a graphene hybrid plasmon waveguide for possible single mode operation. The VFEM analysis of the proposed waveguide shows that the undesired SPP modes are extinguished at the output port of the GHPW due to the tunable high optical absorption of the graphene layers. On the other hand, our design contains a hybrid waveguide structure containing two Si nanowires around the central Au nanowire, which significantly enhances the figure of merit of the desired y-polarized dipole mode. Moreover, the GHPW offers high extinction ratios of the undesired SPP modes along with a high figure of merit of the desired SPP mode for a broad range of operating wavelengths containing the telecommunication wavelengths. Also, the extinction ratio of the monopole SPP mode remains moderately high (∼ 30 dB to 40 dB) even after experiencing a large reduction for increasing gap widths. On the other hand, the x-polarized dipole mode demonstrates a negligible variation for a wide range of d₁. Finally, the GHPW still shows a good single-mode waveguiding performance when the optical anisotropy of graphene is considered. Therefore, our GHPW shows a high promise in the design of broadband single mode plasmon waveguide, which could possibly be used in designing MDM nanoplasmonic optical networks.

Acknowledgments

The authors are thankful to Dr. Ahmed Zubair of BUET for his valuable comments on the fabrication of the proposed structure in this work.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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No. of graphene monolayers	ER ₁	ER ₂
4	75.52	6.41
6	93.3	10.75
8	104.6	16.05

References	Structure (mode order)	$F O M_{n o r m} (= \frac{L_{p} / λ}{2 \sqrt{A_{e f f} / (π λ^{2})}})$
Gao et al. [27]	Single graphene-coated nanowire (m = 0 mode)	13.27
Teng et al. [45]	Single graphene-coated elliptical nanowire (m = 1 mode)	250.66
Teng et al. [46]	Graphene-coated double nanowire system (gap supermode)	79.76
Sun et al. [47]	Vertically coupled double graphene-coated nanowires (fundamental supermode)	131
Our design	Graphene-based IMI hybrid nanowire plasmonic waveguide (m = 1 mode)	268.40

No. of graphene monolayers	ER ₁	ER ₂
4	75.52	6.41
6	93.3	10.75
8	104.6	16.05

References	Structure (mode order)	$F O M_{n o r m} (= \frac{L_{p} / λ}{2 \sqrt{A_{e f f} / (π λ^{2})}})$
Gao et al. [27]	Single graphene-coated nanowire (m = 0 mode)	13.27
Teng et al. [45]	Single graphene-coated elliptical nanowire (m = 1 mode)	250.66
Teng et al. [46]	Graphene-coated double nanowire system (gap supermode)	79.76
Sun et al. [47]	Vertically coupled double graphene-coated nanowires (fundamental supermode)	131
Our design	Graphene-based IMI hybrid nanowire plasmonic waveguide (m = 1 mode)	268.40

Design of a broadband single mode hybrid plasmonic waveguide incorporating silicon nanowire

Abstract

1. Introduction

2. Design of a waveguide and analysis method

3. Results and discussions

3.1. SPP modes of a single Au nanowire in SiO₂ cladding

3.2. Modal characteristics of the proposed GHPW for variable gap widths

3.3. Multiwavelength operability of the proposed GHPW

3.4. Effects of graphene’s anisotropy on waveguiding performance

4. Conclusion

Acknowledgments

Disclosures

References

Cited By

Figures (11)

Tables (2)

Equations (12)

Optical Materials Express

Abstract

1. Introduction

2. Design of a waveguide and analysis method

3. Results and discussions

3.1. SPP modes of a single Au nanowire in SiO2 cladding

3.2. Modal characteristics of the proposed GHPW for variable gap widths

3.3. Multiwavelength operability of the proposed GHPW

3.4. Effects of graphene’s anisotropy on waveguiding performance

4. Conclusion

Acknowledgments

Disclosures

References

Cited By

Figures (11)

Tables (2)

Equations (12)

Optical Materials Express

3.1. SPP modes of a single Au nanowire in SiO₂ cladding