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Abstract
In this study, the dispersion equations of a graphene-coated nanowire (GN) are solved. It is found that in this waveguide, besides the surface plasmon polaritons (SPPs), there is another branch of guided modes, called photonic-like modes. The propagation distances of the photonic-like modes can be five orders of magnitude longer than those of the SPPs. Moreover, they can be modulated in the range of 10−4 to 1 m by changing the chemical potential of graphene. In particular, the mode field distributions remain nearly unchanged during the modulation. Based on the analysis performed using COMSOL Multiphysics, we further demonstrated that the propagation losses of the photonic-like modes are dependent on not only the chemical potential of graphene, but also the mode power proportion inside graphene. The photonic-like modes have tremendous potential to be used in optical switches, modulators, and switches in magnetic fields at the nanoscale.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, tunable optoelectronic components have played an increasingly important role in numerous fields, such as optical communication [1], optical signal processing [2], digital imaging [3], and optical interconnection [4]. Graphene has excellent photonic and electronic properties, such as broadband high-speed light–matter interaction [5] and tunable chemical potential [6]. Therefore, it has been found to be extremely useful in constructing tunable subwavelength optoelectronic devices and components, such as directional coupler switches [7] and optical modulators [8]. In subwavelength optics, graphene-coated nanowires (GNs), including isolated GNs [9–13], GN dimers [14–17], and GN arrays [18–20], have been investigated intensively. It is noteworthy that previous theoretical investigations mainly focused on the surface plasmon polaritons (SPPs) in GNs, which can achieve the localization of the electromagnetic wave far beyond the diffraction limit [11,21–25]. However, the SPPs suffer from short propagation distances.
In this study, by solving the dispersion equations, we found that in a GN, besides the SPPs [26,27], there is another branch of guided modes, called photonic-like modes. The photonic-like modes have some properties that are completely different from those of the SPPs. Some examples are the long propagation distances (five orders of magnitude longer than those of the SPPs) and the tunable mode loss with an unchanged field distribution. Based on the mode analysis, it is demonstrated that the GN can guide three types of photonic-like modes: hybrid modes with both longitudinal electric and magnetic fields (EH), transverse magnetic modes (TM), and transverse electric modes (TE). In contrast to the SPPs, the lowest order EH mode is cutoff-free, however, the TM modes possess modal cutoff characteristics. Notably, the TE modes, which do not appear in the SPP branch, have azimuthal electric field distributions and longitudinal magnetic field distributions. Therefore, they can provide an approach to control the switching of the magnetic fields at the GN end and allow adjustable interactions with magnetic targets at the nanoscale. Moreover, the propagation loss of the photonic-like modes can be modulated in the range of 1 to ${10^4}$ ${{\textrm{m}}^{ - 1}}$ (the corresponding propagation distance being in the range of ${10^{\textrm{ - }4}}$ to 1 m) by changing the chemical potential of graphene [11]. However, the effect of the chemical potential on the effective refractive index is negligible, which suggests that the mode field distribution is nearly unchanged during the modulation. According to the analysis using COMSOL Multiphysics, we further demonstrated that the propagation loss of the photonic-like modes is dependent not only on the chemical potential of graphene, but also the mode power proportion inside the graphene. The theoretical results obtained by solving the dispersion equations are highly consistent with the simulations using COMSOL Multiphysics.
2. Theoretical Model
The cross-section of a GN is shown in Fig. 1(a). A dielectric nanowire with a cross-sectional radius, a, and relative permittivity, ${\varepsilon _1}$, is coated by a monolayer graphene, which is exposed to air that has relative permittivity ${\varepsilon _2}{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} 1$. The surface conductivity of graphene, ${\sigma _g}$, can be evaluated using the Kubo formula [28] as follows:
Fig. 1. (a) Schematic of the GN cross-section and coordinate system. (b) Normalized time-averaged Poynting vectors $\left\langle {{S_Z}} \right\rangle$ of the TM, TE, and EH modes, respectively. (c) The amplitudes and polarizations of these modes. Wavelength of the incident light $\lambda {\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} 1.55{\kern 1pt} {\kern 1pt} \mu {\textrm{m}}$.
A Si nanowire with a relative permittivity of ${\varepsilon _1}{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} 11.7$ is adopted, and the cross-sectional radius is $a{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} 250{\kern 1pt} {\kern 1pt} \textrm{nm}$ unless otherwise stated. Using COMSOL Multiphysics, we calculated the normalized time-averaged Poynting vectors $\left\langle {{S_Z}} \right\rangle$ of the TM, TE, and EH modes, as shown in Fig. 1(b). Further, we found that their mode field areas were much larger than those of the SPPs [11]. Both the electric field amplitudes and polarization directions of these modes were also calculated, as show in Fig. 1(c). The field distributions of the photonic-like modes are significantly different from those of the SPPs, because their light guide mechanisms are totally different. The SPPs are caused by the coupling of electromagnetic waves and free electrons on the surface of the graphene, but the photonic-like modes are formed by the total internal reflection, which makes their field distributions similar to those of the conventional photonic modes in a bare nanowire [29].
The propagation characteristics of the photonic-like modes can be quantitatively described by the dispersion equation as follows [11]:
Equations (5) and (6) are the dispersion equations of the TM and TE modes, respectively. $m \ne 0$ corresponds to the EH modes. Based on Eq. (4), the dispersion equation of the lowest order EH mode can be rewritten as
3. Results and discussion
Based on Eqs. (5)–(7), we can calculate the effective refractive index, ${n_{eff}}$, and mode loss, ${\beta _2}$, of the TM, TE, and EH modes, respectively, where ${n_{eff}}{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} {{{\beta _1}} \mathord{\left/ {\vphantom {{{\beta_1}} {{\textrm{k}_0}}}} \right.} {{\textrm{k}_0}}}$. The dependences of ${n_{eff}}$ and ${\beta _2}$ on the chemical potential of graphene, ${\mu _c}$, are shown as solid lines in Figs. 2(a) and 2(b), respectively. In Fig. 2(a), we can observe that ${n_{eff}}$ is almost constant with the variation in ${\mu _c}$; therefore, the effect of graphene on ${n_{eff}}$ is negligible, which indicates that the mode field distributions are nearly unchanged with the change in ${\mu _c}$ in this range. Figure 2(b) shows that ${\beta _2}$ decreases rapidly from the order of magnitude of ${10^4}$ to 1 ${{\textrm{m}}^{ - 1}}$ when ${\mu _c}$ increases from 0.3 to 0.5 eV, whereas ${\beta _2}$ is almost constant when ${\mu _c}$ changes in other ranges. Moreover, the changing trend of ${\beta _2}$ with ${\mu _c}$ is nearly the same as that of the real part of the graphene surface conductivity, ${\mathop{Re}\nolimits} [{{\sigma_g}} ]$, with ${\mu _c}$ [30]. Because ${\mathop{Re}\nolimits} [{{\sigma_g}} ]$ is proportional to the absorption of graphene, the mode loss is mainly caused by graphene. Therefore, by changing ${\mu _c}$ via external electric fields [31], chemical doping, and temperature, the mode loss of the photonic-like modes can be modulated in the range of 1–${10^4}$ m−1 (corresponding to the propagation distance, L, in the range of 1–${10^{\textrm{ - }4}}$ m according to $L{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} {1 \mathord{\left/ {\vphantom {1 {{\beta_2}}}} \right.} {{\beta _2}}}$). However, simultaneously, the mode field distribution is nearly unchanged. When the length of the GN was appropriately selected, the TM, TE, and EH modes could be used for optical switches or modulators operating in different polarization states. In particular, the TE mode could also be used to control the switching of magnetic fields at the GN end and provide adjustable interaction with magnetic targets at the nanoscale, owing to its longitudinal magnetic field distribution. Moreover, we also numerically calculated the above mode characteristics using COMSOL Multiphysics, as shown in the form of dots, triangles, and rectangles on the corresponding solid lines in Figs. 2(a) and 2(b), respectively. The results obtained using the two methods are highly consistent.
Fig. 2. Dependences of (a) effective refractive index and (b) mode loss on the chemical potential of graphene. The solid lines show the calculation results obtained by solving the dispersion equations. The dots, triangles, and rectangles on the solid lines are the calculation results obtained using COMSOL. (The above scheme is also used in Figs. 3–5).
We further calculated the dependences of ${n_{eff}}$ and ${\beta _2}$ on the relative permittivity of the nanowire, ${\varepsilon _1}$, respectively, which are shown as solid lines in Figs. 3(a) and 3(b). With the increase in ${\varepsilon _1}$, the mode power proportion inside the nanowire having a large refractive index increases, and so, ${n_{eff}}$ also increases. In contrast to the SPPs, there are cutoff points (near ${\varepsilon _1}{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} 7$) for both the TM and TE modes, because the total internal reflection condition fails to hold for ${\varepsilon _1}{\kern 1pt} {\kern 1pt} < {\kern 1pt} {\kern 1pt} 7$. The cutoff characteristics are very similar to those of the conventional photonic modes in a bare nanowire. However, the lowest-order EH mode is cutoff-free, and ${n_{eff}}$ asymptotically approaches 1 as ${\varepsilon _1}$ tends to 1. In addition, the ${n_{eff}}$ values of the guided TM, TE, and EH modes always satisfy $1{\kern 1pt} {\kern 1pt} < {\kern 1pt} {\kern 1pt} {n_{eff}}{\kern 1pt} {\kern 1pt} < {\kern 1pt} {\kern 1pt} \sqrt {{\varepsilon _1}}$. Figure 3(b) indicates that as ${\varepsilon _1}$ increases, first ${\beta _2}$ increases and then decreases. Although the absorption coefficient of graphene is constant, the mode field distributions change with ${\varepsilon _1}$, which leads to a change in the mode power proportion inside graphene. Thus, the mode losses change with ${\varepsilon _1}$. The simulated results obtained using COMSOL Multiphysics are shown as dots, triangles, and rectangles on the corresponding solid lines in Fig. 3, respectively. Furthermore, we use $\xi {\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} {{{P_g}} \mathord{\left/ {\vphantom {{{P_g}} {{P_t}}}} \right.} {{P_t}}}$ to describe the mode power proportion inside graphene, where ${P_\textrm{g}}{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} \int\!\!\!\int {\left\langle {{S_z}} \right\rangle } {\kern 1pt} {\kern 1pt} d{S_g}$ (${S_g}$ corresponding to a region of graphene on the cross-section) and ${P_\textrm{t}}{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} \int\!\!\!\int {\left\langle {{S_z}} \right\rangle } {\kern 1pt} {\kern 1pt} d{S_t}$ (${S_t}$ corresponding to the entire cross-section of the waveguide). The dependences of $\xi$ on ${\varepsilon _1}$ are shown in Fig. 3(c). It can be observed that for all the three modes, the changing trends of ${\beta _2}$ with ${\varepsilon _1}$ are consistent with those of $\xi$ with ${\varepsilon _1}$. This indicates that the propagation losses of the photonic-like modes are highly dependent on the mode power proportion inside graphene, in addition to the graphene chemical potential.
Fig. 3. Dependences of (a) effective refractive index, (b) mode loss, and (c) mode power proportion inside graphene on the relative permittivity of the nanowire.
The cross-sectional radius of the nanowire, a, also affects the propagation characteristics of the photonic-like modes, and the dependences of ${n_{eff}}$ and ${\beta _2}$ on a are shown as lines in Figs. 4(a) and 4(b), respectively. When a increases, the power proportion inside the nanowire increases for all the three modes, so that ${n_{eff}}$ increases and finally approaches $\sqrt {{\varepsilon _1}} {\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} 3.42$. It is observed from Fig. 4(a) that the cutoff points are near $a{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} 180{\kern 1pt} {\kern 1pt}$ nm for the TM and TE modes, however, the lowest-order EH mode is cutoff-free. Although the absorption coefficient of graphene is constant here, ${\beta _2}$ first increases and then decreases as a increases, as shown in Fig. 4(b), which mainly results from the change in the mode power proportion inside graphene $\xi$.
Fig. 4. Dependences of (a) effective refractive index and (b) mode loss on the cross-sectional radius of the nanowire.
When the wavelength, $\lambda$, increases, the mode power proportion in air having a small refractive index increases, and so, ${n_{eff}}$ decreases, as shown in Fig. 5(a). The cutoff points of the TM and TE modes are at $\lambda = {\kern 1pt} 2.0{\kern 1pt} $µm and $\lambda = {\kern 1pt} 2.1{\kern 1pt}$µm, respectively, and the lowest-order EH mode is still cutoff-free. Because the propagation loss of the photonic-like modes is dependent not only on the real part of the surface conductivity of graphene, ${\mathop{Re}\nolimits} [{{\sigma_g}} ]$, but also on the power proportion inside graphene $\xi$; the dependence of ${\beta _2}$ on $\lambda$, as shown in Fig. 5(b), is more complex than that on the chemical potential, ${\mu _c}$, the relative permittivity of the nanowire, ${\varepsilon _1}$, or the nanowire radius, a. This is because both ${\mathop{Re}\nolimits} [{{\sigma_g}} ]$ and $\xi$ are dependent on $\lambda$.
4. Summary
To conclude, by solving the dispersion equations, we found that in a GN, there is a branch of guided modes in addition to the SPPs whose propagation distance is five orders of magnitude longer than those of the SPPs. Based on the mode analysis, it is demonstrated that the GN can guide three types of photonic-like modes, where the lowest-order EH mode is cutoff-free; however, the TM and TE modes have modal cutoff characteristics. Moreover, the propagation losses of the photonic-like modes are dependent on both the real part of the surface conductivity of graphene and the mode power proportion inside graphene. The mode loss can be modulated in the range of 1–${10^4}$ ${{\textrm{m}}^{ - 1}}$ (corresponding to the propagation distance, L, being in the range of ${10^{\textrm{ - }4}}$ to 1 m) by changing the surface conductivity of graphene via the chemical potential. Both the propagation loss and mode field distribution of the photonic-like modes are dependent on the relative permittivity of the nanowire, ${\varepsilon _1}$, nanowire radius, a, and wavelength of the incident light, $\lambda$. However, the chemical potential of graphene, ${\mu _c}$, affects only the propagation loss, and not the mode field distribution. The results obtained may motivate a new class of applications in optical switches, modulators, and switches in magnetic fields at the nanoscale.
Funding
National Natural Science Foundation of China (11874270); Fund Project for Shenzhen Fundamental Research Program (JCYJ20170818143652693, JCYJ20180305125000525).
Disclosures
The authors declare no conflicts of interest.
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