Optical bistability in graphene-wrapped dielectric nanowires

K. Zhang; L. Gao

doi:10.1364/OE.25.013747

1. Introduction

Nonlinear optical effects play an important role in modern photonic functionalities, including ultrafast optical switching, optical transistors, optical modulation and so on [1]. However, governed by photon–photon interactions, optical nonlinearities are inherently weak. In general, optical nonlinearities are superlinearly field-dependent and may be enhanced with plasmon resonant mechanisms. For instance, plasmonic-enhanced-nonlinear structures such as metal/dielectric composites have been widely studied [2–4]. Such composites support surface plasmonic resonances in the interface of the dielectric and metal [5–7], resulting in strong electromagnetic field and enhanced optical nonlinearity. Such enhancement of the optical nonlinearity may be helpful for dramatically shortening the response time and allowing nonlinear optical components to be scaled down in size [8].

It is well-known that in a nonlinear system, due to the self-feedback mechanism, optical bistable states may occur. Optical bistability (OB) is a way of controlling light with light [9, 10], where a nonlinear optical system shows two different values of the local field for one incident field. In this connection, bistable transmission in a Fabry-Perot interferometer filled with a nonlinear medium [11] and hysteretic reflection at a nonlinear interface [12], were investigated. In addition, some researchers proposed nonlinear plasmonic cloaks [13] to realize all-optical scattering nanoswitches. Moreover, the variational approach [14], self-consistent mean-field approximation in conjunction with the spectral representation method [15, 16], and nonlinear Mie theory [17, 18] were separately developed for the study of the optical bistable behavior in nonlinear plasmonic nanocomposites.

On the other hand, graphene, as an excellent optoelectronic material [19], exhibits an intrinsic high nonlinear optical response [20, 21] and extremely large electron mobilities [22, 23] in several frequency regions. The nonlinearity of graphene was explored to be quite useful for the potential applications both theoretically and experimentally, such as the mode-locking fiber [24], harmonic generations [25, 26], nonlinear surface plasmons (SPs) [27, 28] and so on. Besides, optical bistable behavior of graphene/graphene-based structures have been widely investigated in one-dimensional infinitely extended system [29–32]. Similarly, in nanostructured graphene, such as nanoribbons, nanowires, and nanospheres, surface plasmonic bistability [33], plasmonic mode bifurcation [34], and bistable transmittance [35] at low input power were proposed, which offer many insights into the role of nonlinear interaction in nanoscale graphene.

In this paper, we study the optical bistability of the near-field and far-field spectra in the graphene-wrapped nanocylinders theoretically. This kind of microstructures may be realized in experiments [36], and has been demonstrated for optical cloaking [37], plasmonic waveguide [38], and superscattering [39]. Since the local fields within the graphene are generally inhomogeneous, we shall adopt the self-consistent mean-field approximation [15, 16] in conjunction with full-wave electromagnetic scattering theory [37–40] to solve the local electromagnetic fields as a function of the applied field. When the radius of the dielectric nanocylinders or nanowires is far smaller than the incident wavelength, i.e., in the quasistatic limit, we take one step forward to derive the formulae for the local electric fields. We shall show that when the applied field is small, both methods yield same results in the quasistatic limit. However, when the applied field is further enhanced, one observes two optical bistable regions with the nonlinear full-wave electromagnetic theory, which is omitted by the quasistatic theory. Furthermore, the threshold fields of the single and two OBs are tunable either by varying the sizes, permittivity of cylinders, or changing the chemical potential. These results promise the graphene-wrapped cylinders candidates for all-optical switching, which has potential applications in optical communications and optical computing.

2. Theoretical model and methods

We start our work by considering the two-dimensional (2D) system, as shown in Fig. 1, where a transverse magnetic (TM) -polarized plane wave is applied on the graphene-wrapped nanocylinders or nanowires with radius a. The relative permittivity of the dielectric nanowire and the host medium is $ε$ and $ε_{h}$ respectively. Since the single layer graphene is only one-atom thick, much smaller than the cylinder size, the graphene coating can be theoretically characterized as a two-dimensional homogenized conducting film with Kerr-like nonlinear surface conductivity [26, 32] $σ_{g} = σ_{0} + σ_{3} {| E |}^{2}$ , where $σ_{0}$ and $σ_{3}$ correspond to the linear term and third order nonlinear term of the graphene conductivity, and $E$ is the local field within the graphene monolayers.

Fig. 1 Schematic diagram of the model

Download Full Size | PDF

A. Linear theories for linear graphene-wrapped cylinders

A.1 Derivations based on full-wave electromagnetic scattering theory (FWST))

As illustrated in Fig. 1, the incident electric field is perpendicular to the xz plane and spreads along $x$ axis with the form of $E = \hat{y} E_{0} e^{i k x} \cdot e^{- i ω t}$ , where $k = k_{0} \sqrt{ε_{h}}$ denotes the wave number in surrounding medium. Based on the full wave scattering theory, the general solutions for the local electromagnetic field can be written as [37],

\begin{array}{l} E_{i} = - \frac{i}{k} \sum_{n = - \infty}^{\infty} E_{n} M_{n}^{(1)}, H_{i} = \frac{- 1}{ω μ} \sum_{n = - \infty}^{\infty} E_{n} N_{n}^{(1)} \\ E_{s} = \frac{i}{k} \sum_{n = - \infty}^{\infty} A_{n} E_{n} M_{n}^{(3)}, H_{s} = \frac{1}{ω μ} \sum_{n = - \infty}^{\infty} A_{n} E_{n} N_{n}^{(3)}, \\ E_{c} = - \frac{i}{k_{1}} \sum_{n = - \infty}^{\infty} F_{n} E_{n} M_{n}^{(1)}, H_{c} = \frac{- 1}{ω μ_{1}} \sum_{n = - \infty}^{\infty} F_{n} E_{n} N_{n}^{(1)} \end{array}

with

k_{1} = k_{0} \sqrt{ε}

and

E_{n} = E_{0} {(- i)}^{n}

. Here

M_{n}

and

N_{n}

are vector cylindrical harmonics, and the upper indices (1) and (3) represent the Bessel function

J_{n}

and the first kind of Hankel function

H_{n}

. In addition,

E_{i}

,

E_{s}

,

E_{c}

together with

H_{i}

,

H_{s}

,

H_{c}

are relevant electric/magnetic incident, scattering and core fields.

For such a nanowire with linear surface conductivity $σ_{0}$ [30, 31], we have the boundary conditions,

\hat{n} \cdot (E_{i} + E_{s} - E_{c}) = 0, \hat{n} \times (H_{c} - H_{i} - H_{s}) = σ_{0} E_{t} .

Applying the boundary conditions at r = a yields,

\begin{array}{l} A_{n} = \frac{J_{n} (x) J_{n}^{'} (m x) - m J_{n}^{'} (x) J_{n} (m x) - i σ_{0} α J_{n}^{'} (x) J_{n}^{'} (m x)}{H_{n} (x) J_{n}^{'} (m x) - m H_{n}^{'} (x) J_{n} (m x) - i σ_{0} α H_{n}^{'} (x) J_{n}^{'} (m x)} \\ F_{n} = \frac{H_{n} (x) J_{n}^{'} (x) - H_{n}^{'} (x) J_{n} (x)}{H_{n} (x) J_{n}^{'} (m x) - m H_{n}^{'} (x) J_{n} (m x) - i σ_{0} α H_{n}^{'} (x) J_{n}^{'} (m x)}, \end{array}

with

α = \sqrt{μ_{0} / ε_{0} ε_{h}}

,

m = \sqrt{ε / ε_{h}}

and

x = k a

.

Then, the distribution of the local electric fields within the dielectric cylinder can be obtained by substituting Eq. (3) into Eq. (1). Especially, when we take r = a, we can achieve the square of modulus of the local fields near the dielectric-graphene interface,

\begin{array}{l} E_{c, l o c a l} = - E_{0} \sum_{n = - \infty}^{\infty} i^{n + 1} F_{n} (i n \frac{J_{n} (k_{1} a)}{k_{1} a} {\hat{e}}_{r} - J_{n}^{'} (k_{1} a) {\hat{e}}_{φ}) e^{i n φ} \\ {| E_{c, l o c a l} |}^{2} = {| E_{0} |}^{2} {| \sum_{n = - \infty}^{\infty} i^{n + 1} F_{n} (i n \frac{J_{n} (k_{1} a)}{k_{1} a} {\hat{e}}_{r} - J_{n}^{'} (k_{1} a) {\hat{e}}_{φ}) e^{i n φ} |}^{2}, \end{array}

together with the square of modulus of the tangential local field within the graphene thin layer

{| E_{l i n, g} |}^{2} = {| E_{0} |}^{2} {| \sum_{n = - \infty}^{\infty} i^{n + 1} F_{n} J_{n}^{'} (k_{1} a) {\hat{e}}_{φ} e^{i n φ} |}^{2} .

Besides, the efficiencies for scattering and extinction $Q_{s c a}$ and $Q_{e x t}$ are expressed as,

Q_{s c a} = \frac{2}{x} ({| A_{0} |}^{2} + 2 \sum_{n = 1}^{\infty} {| A_{n} |}^{2}), Q_{e x t} = \frac{2}{x} Re (A_{0} + 2 \sum_{n = 1}^{\infty} A_{n}),

A.2 Derivations in the quasistatic limit (QL)

Since the diameters of the nanocylinder we employed in our work are much smaller than the incident wavelength, we also put out the derivations in the quasistatic limit for comparison. In this connection, we can involve the electric potentials both inside ( $ϕ_{c}$ ) and outside ( $ϕ_{h}$ ) the cylinders, which satisfy the Laplace equation: $\nabla^{2} ϕ_{c, h} = 0$ [26]. They have the general solutions,

\begin{array}{l} ϕ_{c} = - B E_{0} r \cos φ, \\ ϕ_{h} = (- E_{0} r + \frac{C E_{0}}{r}) \cos φ . \end{array}

To solve the coefficients B and C, we adopt the following boundary conditions [26, 35],

\hat{n} \times [E_{h} - E_{c}] |_{r = a} = 0, \hat{n} \cdot [D_{h} - D_{c}] |_{r = a} = ρ,

where

E_{c, h} = - \nabla ϕ_{c, h}

and

D_{c, h} = ε_{c, h} E_{c, h}

(

ε_{c} = ε

) are the electric field and electric displacement vector inside and outside the cylinder. The symbol

ρ

represents the surface charges, which has the relation

i ω ρ = \nabla_{s} \cdot J

with the surface current density

J

, and the operator

\nabla_{s}

stands for the surface divergence. Combining Eq. (7) with Eq. (8), we have

B = \frac{2 ε_{h}}{ε + ε_{h} + γ}, C = \frac{a^{2} (ε - ε_{h} + γ)}{ε + ε_{h} + γ},

with

γ = i σ_{g} / (ω a ε_{0})

.

Based on $E_{c} = - \nabla ϕ_{c} = B E_{0} (\cos φ {\hat{e}}_{r} - \sin φ {\hat{e}}_{φ})$ , one yields

{| E_{l i n, g} |}_{Q L}^{2} = {| E_{c} |}^{2} = {| B |}^{2} {| E_{0} |}^{2},

which indicates the relation between the linear local fields in graphene and the incident field.

In the quasistatic limit, the scattering and extinction efficiencies $Q_{s c a, Q L}$ and $Q_{e x t, Q L}$ can be simplified as [40],

Q_{s c a, Q L} = \frac{π^{2} {(k a)}^{3}}{4} {| \frac{C}{a^{2}} |}^{2}, Q_{e x t, Q L} = π (k a) \frac{Im (C)}{a^{2}} .

B. Nonlinear theories for nonlinear graphene-wrapped cylinders

In this section, we would like to take one step forward to consider the intrinsic nonlinear property of the graphene monolayer. Within the random-phase approximation, the surface conductivity of graphene is written as $σ_{g} = σ_{0} + σ_{3} {| E_{t} |}^{2}$ ,where $σ_{0} = i e^{2} μ_{c} / [π ℏ^{2} (ω + i / τ)]$ , $σ_{3} = - i 9 e^{4} v_{F}^{2} / (8 π μ_{c} ℏ^{2} ω^{3})$ with $e$ , $μ_{c}$ , $ℏ$ , $τ$ and $v_{F}$ being the charge of electron, chemical potential of graphene, reduced Planck constant, electron-phonon relaxation time, and Fermi velocity of electrons. In addition, the explicit form of ${| E_{t} |}^{2}$ in the nonlinear case is assumed to be the form of ${| E_{l i n, g} |}^{2}$ [see Eq. (5)] in the FWST or the one ${| E_{l i n, g} |}_{Q L}^{2}$ [Eq. (10)] in QL [41, 42]. By replacing the linear conductivity $σ_{0}$ in the linear derivations by the nonlinear one $σ_{g}$ , one yields the nonlinear solutions for the nonlinear system. On the other hand, when the nonlinear conductivity of the coated graphene is taken into account, the tangential field would definitely be nonlinear, so ${| E_{t} |}^{2}$ would be replaced by the nonlinear one, namely, in the FWST ${\tilde{σ}}_{g} \approx σ_{0} + σ_{3} {| E_{n o n, g} |}_{F W S T}^{2}$ , with

\begin{array}{l} {| E_{n o n, g} |}_{F W S T}^{2} = {| E_{0} |}^{2} {| \sum_{n = - \infty}^{\infty} i^{n + 1} {\tilde{F}}_{n} J_{n}^{'} (k_{1} a) {\hat{e}}_{φ} e^{i n φ} |}^{2}, \\ {\tilde{F}}_{n} = \frac{H_{n} (x) J_{n}^{'} (x) - H_{n}^{'} (x) J_{n} (x)}{H_{n} (x) J_{n}^{'} (m x) - m H_{n}^{'} (x) J_{n} (m x) - i {\tilde{σ}}_{g} α H_{n}^{'} (x) J_{n}^{'} (m x)}, \end{array}

One the other hand, in the QL ${\tilde{σ}}_{g} \approx σ_{0} + σ_{3} {| E_{n o n, g} |}_{Q L}^{2}$ , with

{| E_{n o n, g} |}_{Q L}^{2} = {| \tilde{B} |}^{2} {| E_{0} |}^{2}, \tilde{B} = \frac{2 ε_{h}}{ε + ε_{h} + i {\tilde{σ}}_{g} / (ω a ε_{0})} .

Equation (12) and Eq. (13) can be readily solved in a self-consistent manner for

{| E_{n o n, g} |}_{F W S T}^{2}

or

{| E_{n o n, g} |}_{Q L}^{2}

as a function of

{| E_{0} |}^{2}

, and hence the desired optical bistable behavior in the near-field may be observed.

In the case that $ε$ and $ε_{h}$ are pure dielectric and have no dissipation, we can alternatively rewrite Eq. (13) as

y (x) = x P^{2} [x^{2} + 2 Q x + Q^{2} + R^{2}],

by defining new dimensionless variables

y = \frac{e^{2} v_{F}^{2}}{ω^{2} μ_{c}^{2}} {| E_{0} |}^{2}

,

x = \frac{e^{2} v_{F}^{2}}{ω^{2} μ_{c}^{2}} {| E_{n o n, g} |}_{Q L}^{2}

, with

P = \frac{1}{2 ε_{h}} \frac{9}{8} \frac{e^{2} E_{F}}{π ℏ^{2} ω^{2} a ε_{0}}

,

Q = (ε + ε_{h}) \frac{8}{9} \frac{π ℏ^{2} ω^{2} a ε_{0}}{e^{2} E_{F}} - \frac{8}{9} \frac{{(ω τ)}^{2}}{{(ω τ)}^{2} + 1}

and

R = \frac{8}{9} \frac{ω τ}{{(ω τ)}^{2} + 1}

.

Besides, the nonlinear far-field spectra such as the nonlinear scattering and extinction efficiencies [Eq. (6) and Eq. (11)] should be modified as,

{\tilde{Q}}_{s c a} = \frac{2}{x} ({| {\tilde{A}}_{0} |}^{2} + 2 \sum_{n = 1}^{\infty} {| {\tilde{A}}_{n} |}^{2}), {\tilde{Q}}_{e x t} = \frac{2}{x} Re ({\tilde{A}}_{0} + 2 \sum_{n = 1}^{\infty} {\tilde{A}}_{n})

and for QL, they are

{\tilde{Q}}_{s c a, Q L} = \frac{π^{2} {(k a)}^{3}}{4} {| \frac{\tilde{C}}{a^{2}} |}^{2}, {\tilde{Q}}_{e x t, Q L} = π (k a) \frac{Im (\tilde{C})}{a^{2}} .

3. Numerical results and discussion

We are now in a position to present some numerical results. For numerical calculations, without loss of generality, we consider a graphene-wrapped dielectric nanowire embedded in pure dielectric medium with relative dielectric constant $ε_{h} = 2.25$ . In addition, the relative permeability of the medium inside and outside the cylinder is taken to be 1.

At first, we investigate the linear scattering efficiency of the graphene coated dielectric cylinder based on Eq. (6) with n = 1. For linear case, we ignore the field-dependent term in Eq. (12) and the surface conductivity $σ_{0}$ is completely complex with its imaginary part $Im (σ_{g}) > 0$ . This indicates the graphene is just a “metallic” thin layer in the terahertz region [43]. As a consequence, the surface-plasmon enhanced scattering efficiencies are observed in Fig. 2. Take a close look at Fig. 2, we conclude that the surface-plasmon resonant wavelength varies with the size, the core permittivity and chemical potential of graphene. In detail, for a fixed chemical potential, the resonant wavelength undergoes a red shift with increasing the cylindrical sizes, accompanied by large enhancement of the resonant peak [see Fig. 2(a)]. This is due to the fact that the graphene layer, as a metal-like metamaterial, its surface-plasmon resonant wavelength tends to be size-dependent. What’s more, by increasing the chemical potential (it can be tuned with the density of the charge carriers through the external electrical gating field and/or chemical doping), one can also achieve large enhancement of the linear scattering efficiency [see Fig. 2(b) or 2(c)]. With the increase of the chemical potential from 0.1 eV to 0.3 eV, the resonant magnitude of the scattering efficiency increases together with a blue shift of the resonant wavelength due to surface plasmon resonance. Similar phenomena were found in [32]. On the other hand, by comparing Fig. 2(b) with Fig. 2(c), we find that for fixed chemical potential, increasing the core permittivity will result in the decrease of the resonant magnitude and the red-shift of the resonant wavelength.

Fig. 2 The linear scattering efficiency for various incident wavelength, chemical potential, and nanowire radius with parameters (a) $ε=2 .25$ , $μ_{c} = 0.3 e V$ , (b) $ε=1 .5$ , $a = 100 n m$ , and (c) $ε=2 .25$ , $a = 100 n m$ . Other parameters are $ε_{h} =2 .25$ and $τ=0 .1 p s$

Download Full Size | PDF

Next, we consider the nonlinear graphene-wrapped dielectric nanowires, with the surface conductivity of graphene being a Kerr-like one. Based on Eq. (12), we analyze the relation between the modulus of nonlinear local field inside graphene $E_{non,g} = \sqrt{{| E_{n o n, g} |}^{2}}$ and the incident electric field $E_{0}$ with incident wavelength $λ = 25 μ m$ , and the results are shown in Fig. 3. Take $μ_{c} = 0.3 e V$ as an example, as the incident field is increased to $3 \times 10^{6} (v / m)$ , the nonlinear field within the graphene jumps discontinuously from $0.75 \times 10^{7} (v / m)$ to $1.4 \times 10^{7} (v / m)$ , and then increases monotonically with further increasing $E_{0}$ ; On the contrary, as one decreases the incident field to the switching-down threshold field $1.2 \times 10^{6} (v / m)$ , the nonlinear field within the graphene monolayer decreases from $1.2 \times 10^{7} (v / m)$ to $0.5 \times 10^{7} (v / m)$ directly [see Fig. 3(a)]. All these properties characterize well the optical bistable behavior. Moreover, we find that both the switching-up and switching-down threshold fields increase with increasing the chemical potential. As a matter of fact, the third-order nonlinear term $σ_{3}$ decreases as the chemical potential increases, and hence the incident field should increase to achieve the hysteresis spectra, so as to keep the nonlinear local field enough large to make the term $σ_{3} {| E_{n o n, g} |}^{2}$ comparable to the linear term. Figure 3(b) and Fig. 3(c) show the near-field spectra for various core permittivity and various sizes. It is evident that both increasing the core permittivity and the sizes are helpful to decrease the threshold fields, which were already predicted but for graphene-wrapped nanospheres. In Fig. 3(c), for comparison, both the results for FWST and for QL are plotted. Good agreement is found, which demonstrate both two theories are adoptable to investigate the bistable behavior within our parameters. Besides, the phenomena that the switching-up and switching-down threshold fields decrease with increasing the size can be understood as follows: increasing the cylinder size results in the enhancement of the linear field within the graphene monolayer [35]. And with such a field enhancement, one would expect a low incident field to realize the optical bistability, and hence a small threshold field. This study provides us one possible way to realize nonlinear nanoswitch devices, whose threshold fields can be tunable with changing the dimension of the cylinders. In view of possible technological applications, these findings are expected to be very useful.

Fig. 3 The modulus of the nonlinear local field $E_{non,g}$ as a function of the incident field $E_{0}$ for varied (a) chemical potentials; (b) sizes and (c) permittivity. Other parameters are $ε_{h} =2 .25$ and $τ=0 .1ps$ .

Download Full Size | PDF

At the same time, we consider the situation of transverse electric (TE) wave incidence, as shown in Fig. 4. It is evident that the curves of the modulus of the nonlinear local field as a function of the external applied field exhibit optical bistability. Different from the TM case, the bistable curves are weakly dependent on the core permittivity and the radius of cylinder. In addition, the threshold fields for TE case are almost 10 times larger than those for the TM case. Actually, for TE incident wave, the direction of electric field is parallel to the axis of cylinder, and hence the local electric field in graphene is also along the axis of cylinder [44]. In this regards, the magnitude of the local electric fields is mainly determined by the material property of graphene instead of the dielectric cylinder, and is difficult to be enhanced through the surface plasmon resonance. As a consequence, one should strengthen the incident electric field to realize the optical bistability. These results and analyses suggest that TM wave is much more appropriate for studying the plasmon-resonance enhanced nonlinearity of the graphene. Therefore, in what follows, we only consider the TM case.

Fig. 4 Same as those in Fig. 3, but for TE case.

Download Full Size | PDF

In Fig. 5, we show $E_{n o n, g}$ and the far-field scattering and extinction efficiencies as a function of the incident wavelength with $E_{0} = 5 \times 1 0^{6} (v / m)$ . Bistable field as a function of the incident wavelength is indeed found, and the bistable region is relatively broad and can be a few micrometers for large chemical potentials, as shown in Fig. 5(a). It is further found that for the same parameters, the maximal value of nonlinear scattering efficiency is almost 2 percent of the resonant peak for linear scattering efficiency [45], with slightly shifting of the nonlinear resonant wavelength [see Fig. 5(b) and Fig. 2]. It suggests a different energy transfer mechanism when one introduces the nonlinearity of graphene in this nanostructure. In addition, a loop is formed due to the existence of the optical bistability above the nonlinear surface-plasmon resonant wavelength. Moreover, both the nonlinear resonant wavelength and the loop exhibit blue-shift with the increasing of the chemical potential. For the nonlinear extinction spectra at $μ_{c} = 0.4 e V$ [see Fig. 5(c)], one observes the bistable extinction again, with an abrupt switching effect at the designed wavelength $λ = 24.2 μ m$ , at which the extinction efficiency can jump from large extinction to almost zero extinction. Note that the values of the extinction efficiency is much larger than the scattering efficiency, showing a good absorption property of the graphene coated cylinder, which promise such structure a candidate for optical absorber. Therefore, our proposed graphene-wrapped nanowires may be the nonlinear nanoswitch device, whose switching frequency is tunable via varying the density of the charge carriers and the incident field. The tunability, combined with the switching properties in the terahertz region, may be helpful for the design of nonlinear nanocircuit components, nanoabsorbers, and new-generation tunable sensors [13].

Fig. 5 (a) Dependence of the modulus of the nonlinear field $E_{n o n, g}$ inside the graphene on the incident wavelength at different chemical potentials; (b) and (c) illustrate the nonlinear far-field spectra versus incident wavelength at different chemical potentials.

Download Full Size | PDF

Next, we consider the nonlinear local field inside the graphene as a function of the incident field within the FWST and the QL. As shown in Fig. 6(a), besides the bistable curve shown in above sections, when the incident field gets stronger, there is another hysteretic curve based on the FWST. However, for QL, one can only predict the bistable behavior in the weak-field case, and the optical bistability in the strong-field region is omitted. Actually, for FWST, both the electric monopole n = 0, dipole n = 1, and high-pole contributions on the optical bistability are included, while only dipole n = 1 contribution is considered in the quasistatic limit. When the strong-field is applied, the equivalent permittivity of the nonlinear graphene wrapped nanowires may become high dielectric, which may induce the electric monopole (or magnetic dipole) with n = 0 [46–48]. Therefore, one understands the discrepancy between the FWST and QL as follows [see Fig. 6(b)]: as we increase the incident electric field i, the contributions from other terms in the expansion functions are negligible compared to the dipole term n = 1, resulting in almost same results; On the contrary, when we further increase the external field, high permittivity takes place, and magnetic dipole n = 0 plays the leading role. Furthermore, it can be proved in Fig. 6 (c) that both two optical bistabilities keep the same tendency with our varying the chemical potential of graphene. We believe our results would offer a thorough understanding in realizing the optical bistability of the graphene wrapped dielectric nanowires.

Fig. 6 The modulus of nonlinear local field $E_{non,g}$ as a function of $E_{0}$ , (a) undethe FWST and QL with $a =100nm$ , $μ_{c} = 0.3 e V$ ; (b) different terms within the FWST;(c) with different chemical potentials. Other parameters are $ε= ε_{h} =2 .25$ .

Download Full Size | PDF

In the end, we consider the lossless case with low incident intensity, where the relaxation time of graphene is infinite in the quasistatic limit. Based on Eq. (14), we study the bistable behavior of the dimensionless field $x$ as a function of $y$ in Fig. 7. To one’s interest, although the chemical potential of graphene changes, the switching-down threshold field is zero. Similar conclusion was made in Ref [35]. for the composite system containing graphene-wrapped spheres. Mathematically, in the lossless case, Eq. (14) is reduced to $y (x) = x P^{2} {(x + Q^{'})}^{2}$ , with $Q^{'} = (ε + ε_{h}) \frac{8}{9} \frac{π ℏ^{2} ω^{2} a ε_{0}}{e^{2} E_{F}} - \frac{8}{9}$ , and the resonant condition $x = - Q^{'}$ leads to $y = 0$ . Besides, the switching threshold fields in our 2D cylindrical structure are about 100 times smaller than those of monolayer graphene [30], indicating it is much easier to realize OB in graphene-wrapped nanocylinders.

Fig. 7 Curves of the dimensionless field x as a function of y,for different values of the chemical potentials.

Download Full Size | PDF

4. Conclusion and discussion

In conclusion, we establish nonlinear full-wave electromagnetic scattering theory and nonlinear quasistatic theory to investigate the near-field and far-field spectra for nonlinear graphene-wrapped dielectric nanowires. Our theory allows us to solve self-consistently the relation between the average field within the nonlinear graphene and the incident field, and to further observe optical bistable behaviors for the near-field, far-field scattering and extinction efficiency. We find that introducing the nonlinearity of graphene results in the decrease in the scattering efficiency and the blue shift of the resonant wavelength. It’s also demonstrated that both theories are adoptable in analyzing the bistable behavior of the graphene-coated nanowires for a relevantly low electric field due to the electric dipole contribution. However, once the incident field is strong enough, the nonlinear full wave scattering theory predicts the other optical bistable region due to the excitation the magnetic dipole, which was omitted within nonlinear quasistatic theory. Furthermore, the up and down threshold fields are highly dependent on the chemical potential of graphene, the radius and the core permittivity. Our study provides alternative degree of freedom to control the local field and scattering (extinction) efficiency with the incident field. We believe that all these novel properties may have great potentials for the design in optoelectronic switching and nano-memories.

Some comments are in order. At first, for our chosen nanosizes around 100nm, the graphene may exhibit nonlocal electromagnetic responses due to its naturally dispersive dielectric response [49]. It would be of great interest to take into account the interaction between the nonlocality and the nonlinearity of the graphene, and to see how the nonlocality or the spatial dispersion affects the optical bistable threshold [50]. Secondly, Our present works may be generalized to investigate the dynamic evolution of the nonlinear polarizations in graphene system, and one may predict the formation of the plasmonic kinks, oscillons, solitons, and so on [51, 52]. The observed bistable properties with the size-tunability and wavelength-tunability of the graphene-wrapped dielectric nanowires in Terahertz region may lead to many interesting applications such as low-field memories, switches, and sensitive tunable sensors.

Funding

National Natural Science Foundation of China (NSFC) (Grant No. 11374223); National Science of Jiangsu Province (Grant No. BK20161210); the Qing Lan project, “333” project (Grant No. BRA2015353); Project Funded by the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.

References and links

1. R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

2. Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010). [CrossRef] [PubMed]

3. T. Hanke, J. Cesar, V. Knittel, A. Trügler, U. Hohenester, A. Leitenstorfer, and R. Bratschitsch, “Tailoring spatiotemporal light confinement in single plasmonic nanoantennas,” Nano Lett. 12(2), 992–996 (2012). [CrossRef] [PubMed]

4. A. Mirzaei, A. E. Miroshnichenko, N. A. Zharova, and I. V. Shadrivov, “Light scattering by nonlinear cylindrical multilayer structures,” J. Opt. Soc. Am. B 31(7), 1595–1599 (2014). [CrossRef]

5. A. K. Sarychev and V. M. Shalaev, “Electromagnetic field fluctuations and optical nonlinearities in metal-dielectric composites,” Phys. Rep. 335(6), 276–371 (2000). [CrossRef]

6. L. Gao, K. W. Yu, Z. Y. Li, and B. Hu, “Effective nonlinear optical properties of metal-dielectric composite media with shape distribution,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036615 (2001). [CrossRef] [PubMed]

7. J. P. Huang and K. W. Yu, “Effective nonlinear optical properties of graded metal-dielectric composite films of anisotropic particles,” J. Opt. Soc. Am. B 22(8), 1640–1647 (2005). [CrossRef]

8. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012). [CrossRef]

9. Y. R. Shen, “Recent advances in optical bistability,” Nature 299(5886), 779–780 (1982). [CrossRef]

10. H. M. Gibbs, Optical bistability: controlling light with light (Academic Press, 1985).

11. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. 36(19), 1135–1138 (1976). [CrossRef]

12. P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a nonlinear interface,” Appl. Phys. Lett. 35(11), 846–848 (1979). [CrossRef]

13. C. Argyropoulos, P. Y. Chen, F. Monticone, G. D’Aguanno, and A. Alù, “Nonlinear plasmonic cloaks to realize giant all-optical scattering switching,” Phys. Rev. Lett. 108(26), 263905 (2012). [CrossRef] [PubMed]

14. D. J. Bergman, O. Levy, and D. Stroud, “Theory of optical bistability in a weakly nonlinear composite medium,” Phys. Rev. B Condens. Matter 49(1), 129–134 (1994). [CrossRef] [PubMed]

15. L. Gao, L. Gu, and Y. Huang, “Effective medium approximation for optical bistability in nonlinear metal-dielectric composites,” Solid State Commun. 129(9), 593–598 (2004). [CrossRef]

16. L. Gao, L. Gu, and Z. Li, “Optical bistability and tristability in nonlinear metal/dielectric composite media of nonspherical particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066601 (2003). [CrossRef] [PubMed]

17. H. Chen, Y. Zhang, B. Zhang, and L. Gao, “Optical bistability in a nonlinear-shell-coated metallic nanoparticle,” Sci. Rep. 6(1), 21741 (2016). [CrossRef] [PubMed]

18. H. L. Chen, D. L. Gao, and L. Gao, “Effective nonlinear optical properties and optical bistability in composite media containing spherical particles with different sizes,” Opt. Express 24(5), 5334–5345 (2016). [CrossRef]

19. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010). [CrossRef]

20. E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105(9), 097401 (2010). [CrossRef] [PubMed]

21. M. M. Jadidi, J. C. König-Otto, S. Winnerl, A. B. Sushkov, H. D. Drew, T. E. Murphy, and M. Mittendorff, “Nonlinear Terahertz absorption of graphene plasmons,” Nano Lett. 16(4), 2734–2738 (2016). [CrossRef] [PubMed]

22. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004). [CrossRef] [PubMed]

23. P. Tassin, T. Koschny, and C. M. Soukoulis, “Graphene for Terahertz applications,” Science 341(6146), 620–621 (2013). [CrossRef] [PubMed]

24. H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express 17(20), 17630–17635 (2009). [CrossRef] [PubMed]

25. S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood, “Optical third-harmonic generation in graphene,” Phys. Rev. X 3(2), 021014 (2013). [CrossRef]

26. D. A. Smirnova, I. V. Shadrivov, A. E. Miroshnichenko, A. I. Smirnov, and Y. S. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 035412 (2014). [CrossRef]

27. J. D. Cox and F. Javier García de Abajo, “Electrically tunable nonlinear plasmonics in graphene nanoislands,” Nat. Commun. 5, 5725 (2014). [CrossRef] [PubMed]

28. R. Yu, J. D. Cox, and F. J. G. de Abajo, “Nonlinear plasmonic sensing with nanographene,” Phys. Rev. Lett. 117(12), 123904 (2016). [CrossRef] [PubMed]

29. X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Sci. Rep. 5(1), 12271 (2015). [CrossRef] [PubMed]

30. N. M. R. Peres, Y. V. Bludov, J. E. Santos, A.-P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90(12), 125425 (2014). [CrossRef]

31. S. Gong, T. Zhao, M. Sanderson, M. Hu, R. B. Zhong, X. X. Chen, P. Zhang, C. Zhang, and S. G. Liu, “Transformation of surface plasmon polaritons to radiation in graphene in terahertz regime,” Appl. Phys. Lett. 106(22), 223107 (2015). [CrossRef]

32. A. Ciattoni and C. Rizza, “Graphene-nonlinearity unleashing at lasing threshold in graphene-assisted cavities,” Phys. Rev. A 91(5), 053833 (2015). [CrossRef]

33. T. Christensen, W. Yan, A. P. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92(12), 121407 (2015). [CrossRef]

34. R. J. Li, X. Lin, S. S. Lin, X. M. Zhang, E. P. Li, and H. S. Chen, “Graphene induced mode bifurcation at low input power,” Carbon 98, 463–467 (2016). [CrossRef]

35. Y. Huang, A. E. Miroshnichenko, and L. Gao, “Low-threshold optical bistability of graphene-wrapped dielectric composite,” Sci. Rep. 6(1), 23354 (2016). [CrossRef] [PubMed]

36. W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014). [CrossRef] [PubMed]

37. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008).

38. P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011). [CrossRef] [PubMed]

39. Y. Gao, G. Ren, B. Zhu, H. Liu, Y. Lian, and S. Jian, “Analytical model for plasmon modes in graphene-coated nanowire,” Opt. Express 22(20), 24322–24331 (2014). [CrossRef] [PubMed]

40. R. J. Li, X. Lin, S. S. Lin, X. Liu, and H. S. Chen, “Tunable deep-subwavelength superscattering using graphene monolayers,” Opt. Lett. 40(8), 1651–1654 (2015). [CrossRef] [PubMed]

41. L. Gao and Z. Y. Li, “Self-consistent formalism for a strongly nonlinear composite: Comparison with variational approach,” Phys. Lett. A 219(5–6), 324–328 (1996). [CrossRef]

42. W. M. V. Wan, H. C. Lee, P. M. Hui, and K. W. Yu, “Mean-field theory of strongly nonlinear random composites: Strong power-law nonlinearity and scaling behavior,” Phys. Rev. B Condens. Matter 54(6), 3946–3953 (1996). [CrossRef] [PubMed]

43. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011). [CrossRef] [PubMed]

44. T.-H. Xiao, L. Gan, and Z.-Y. Li, “Efficient manipulation of graphene absorption by a simple dielectric cylinder,” Opt. Express 23(15), 18975–18987 (2015). [CrossRef] [PubMed]

45. V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, “Size dependent χ(3) for conduction electrons in Ag nanoparticles,” Nano Lett. 4(8), 1535–1539 (2004). [CrossRef]

46. K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102(13), 133901 (2009). [CrossRef] [PubMed]

47. W. Liu, A. E. Miroshnichenko, R. F. Oulton, D. N. Neshev, O. Hess, and Y. S. Kivshar, “Scattering of core-shell nanowires with the interference of electric and magnetic resonances,” Opt. Lett. 38(14), 2621–2624 (2013). [CrossRef] [PubMed]

48. D. Gao, L. Gao, A. Novitsky, H. Chen, and B. Luk’yanchuk, “Topological effects in anisotropy-induced nano-fano resonance of a cylinder,” Opt. Lett. 40(17), 4162–4165 (2015). [CrossRef] [PubMed]

49. A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015). [CrossRef]

50. Y. Huang and L. Gao, “Tunable Fano resonances and enhanced optical bistability in composites of coated cylinders due to nonlocality,” Phys. Rev. B 93(23), 235439 (2016). [CrossRef]

51. R. E. Noskov, P. A. Belov, and Y. S. Kivshar, “Subwavelength modulational instability and plasmon oscillons in nanoparticle arrays,” Phys. Rev. Lett. 108(9), 093901 (2012). [CrossRef] [PubMed]

52. D. A. Smirnova, R. E. Noskov, L. A. Smirnov, and Y. S. Kivshar, “Dissipative plasmon solitons in graphene nanodisk arrays,” Phys. Rev. B 91(7), 075409 (2015). [CrossRef]

Optical bistability in graphene-wrapped dielectric nanowires

Abstract

1. Introduction

2. Theoretical model and methods

A. Linear theories for linear graphene-wrapped cylinders

A.1 Derivations based on full-wave electromagnetic scattering theory (FWST))

A.2 Derivations in the quasistatic limit (QL)

B. Nonlinear theories for nonlinear graphene-wrapped cylinders

3. Numerical results and discussion

4. Conclusion and discussion

Funding

References and links

Cited By

Figures (7)

Equations (16)

Optics Express