Dynamical manipulation of Cosine-Gauss beams in a graphene plasmonic waveguide

Xueqing He; Tigang Ning; Rujiang Li; Li Pei; Jingjing Zheng; Jing Li

doi:10.1364/OE.25.013923

1. Introduction

As the progress of nanotechnology, surface plasmon polaritons (SPPs) have attracted increasing interest in nanophotonics [1]. Due to the applications in particle manipulation [2], self-imaging [3] and light bullets formation [4], various kinds of non-diffracting surface plasmon beams have been proposed, such as Airy beams [5–8], Bessel-like beams [9] and Cosine-Gauss beams (CGBs) [10]. Among these beams, CGBs are the most distinctive ones because of their straight propagation trajectories which can solve the stringent limitations of bending trajectories in applications. CGBs can be generated in experiments using different methods, including the plane wave interference [10,11], grating coupling [12] and using a complex source point [13].

The control of SPPs on metal surfaces is of vital importance and a challenging task in the design of ultra-compact integrated micro/nano optical systems [14–17]. For the CGBs, several methods have been proposed to control their traveling, such as phase control [18], and using complicated groove patterns [19] and near-field plasmonic holograms [20]. However, all of those methods are cumbersome and not dynamical. Thus, searching for a better platform to dynamically manipulate the CGBs is highly imperative. Graphene, as a newly established two dimensional (2D) material, its surface conductivity is almost purely imaginary in the THz and far infrared frequencies, and its chemical potential can be actively tuned by a gate voltage with suitable chemical doping [21–23]. These interesting properties provide the possibility to dynamically manipulate the CGBs in a graphene-based platform.

In this paper, as a demonstration purpose, we numerically study the CGBs can be dynamically manipulated in a graphene-based planar plasmonic waveguide. Due to the tunability of the chemical potential of graphene, the propagation constant of the plasmonic mode can be actively changed. Under a fixed angle, both the transverse oscillation period and the longitudinal propagation length are dynamically manipulated since they are related with the real and imaginary parts of the wave vector, respectively. The graphene-based planar plasmonic waveguide provides a platform with dynamical manipulation to investigate the propagation dynamics of CGBs, as well as other non-diffracting beams.

2. Structure and model

Figure 1 shows the proposed plasmonic waveguide structure, where the 2D graphene monolayer is sandwiched by the dielectric 1 and dielectric 2. The relative permittivities of the two dielectrics are $ε_{1}$ and $ε_{2}$ , respectively. Since graphene’s thickness is extremely small compared with the geometry sizes of the dielectric media, we can treat the graphene monolayer as a conducting film [24–27] and its electromagnetic properties are characterized by a surface conductivity $σ_{g}$ [28, 29], which can be calculated by the Kubo formula under the local random phase approximation (RPA)

σ_{g} = σ_{i n t r a} (ω, T, τ, μ_{c}) + σ_{inter} (ω, T, τ, μ_{c}),

In this formula,

σ_{i n t r a} (ω, T, τ, μ_{c}) = \frac{i e^{2} k_{B} T}{π ℏ^{2} (ω + i τ^{- 1})} [\frac{μ_{c}}{k_{B} T} + 2 \ln (\exp (- \frac{μ_{c}}{k_{B} T}) + 1)]

is the contribution from the intraband electronic transitions, and

σ_{i nter} (ω, T, τ, μ_{c}) = \frac{i e^{2}}{4 π ℏ} \ln (\frac{2 | μ_{c} | - (ω + i τ^{- 1}) ℏ}{2 | μ_{c} | + (ω + i τ^{- 1}) ℏ})

is the contribution from the interband electronic transitions, where the formula for the interband conductivity is valid under the condition

k_{B} T < < | μ_{c} |, ℏ ω

[28]. the intraband and interband contributions rely on the angular frequency

ω

, temperature

T

, relaxation time

τ

and chemical potential

μ_{c}

,

k_{B}

is the Boltzmann's constant,

ℏ = h / 2 π

is the reduced Planck's constant, and -e is the charge of the electron. In our graphene plasmonic waveguide, the chemical potential

μ_{c}

is controlled by the application of a gate voltage with suitable chemical doping [21–23].

Fig. 1 Schematic of the plasmonic waveguide structure, where a graphene monolayer is placed on the x-y plane and sandwiched by the dielectric 1 and dielectric 2.

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In this waveguide, TM plasmonic mode exists [28], where the fields are localized on the graphene surface and decay exponentially in z direction. The electric field component satisfies Helmholtz equation [30]

\nabla^{2} E_{z} (x, y, z) + ε_{d} k_{0}^{2} E_{z} (x, y, z) = 0,

where

k_{0} = ω / c = 2 π f / c

is the wave vector in vacuum,

ε_{d}

is the relative permittivity of the dielectrics. Equation (4) has the following surface wave solution

E_{z} (x, y, z) = {\begin{matrix} A \exp (i k_{x} x) \cos (k_{y} y) \exp (- α z), & z > 0 \\ A \exp (i k_{x} x) \cos (k_{y} y) \exp (α z), & z < 0 \end{matrix},

where

α

is the decay factor and A is an arbitrary constant. This beam is called as a Cosine beam due to the cosine profile in the transverse y direction. It can be decomposed into two propagating SPP waves with an angle

2 θ

, where the components of the wave vector satisfy

k_{y} = k_{s p} \sin θ

and

k_{s p}^{2} = k_{x}^{2} + k_{y}^{2} = α^{2} + ε_{d} k_{0}^{2}

The wave vector of the SPP wave can be calculated from the dispersion relation [23,25]

\frac{k_{2}}{ε_{2}} + \frac{k_{1}}{ε_{1}} + \frac{i σ_{g} η_{0} k_{2} k_{1}}{k_{0} ε_{2} ε_{1}} = 0

where

k_{1} = \sqrt{β^{2} - k_{0}^{2} ε_{1}}

,

k_{2} = \sqrt{β^{2} - k_{0}^{2} ε_{2}}

and

β

is the propagation constant and equal to the wave vector

k_{s p}

, and

η_{0} = \sqrt{μ_{0} / ε_{0}}

is the intrinsic impedance of free space [28].

The Cosine beam is ideally non-diffracting, as it is the exact solution of Eq. (4) and its transverse intensity profile is independent on the propagation distance. But the Cosine beam presented in Eq. (5) carries infinite energy theoretically and cannot be realized in practice. For the sake of practical significance, we can modulate the Cosine beam with a Gaussian envelop, The resulting Cosine-Gauss beam preserves non-diffracting properties in the paraxial approximation and carries a finite energy [10, 31]. The field distribution of a CGB is

E_{z} (x, y, z) = {\begin{matrix} A \exp (i k_{x} x) \cos (k_{y} y) \exp (- \frac{y^{2}}{w_{0}^{2}}) \exp (- α z), & z > 0 \\ A \exp (i k_{x} x) \cos (k_{y} y) \exp (- \frac{y^{2}}{w_{0}^{2}}) \exp (α z), & z < 0 \end{matrix}

where

w_{0}

denotes the beam waist. Compared with the Cosine beam, the CGB is governed by the Cosine-Gauss function in y direction. Strictly speaking, Eq. (9) is no longer an accurate solution of the Helmholtz equation due to the introduction of the Gauss function. However, if the beam waist is large enough, the CGB can still be treated as an approximated solution with the nearly non-diffracting behavior [10]. Its transverse intensity profile is a localized mode and propagates along the x direction without diffractive spreading. It is worth to note that the Cosine-Gauss beam propagates in a linear medium and it is different from the spatial solitons, which commonly appear in nonlinear media.

About excitation of the CGBs, the grating coupler in line grooves [10] and conical mounting configuration [12] are often used. The CGBs excitation includes two steps: the generation of surface plasmon polaritions (SPPs) and the generation of Cosine-Gaussian plasmons. A focused Gaussian beam from a laser can be coupled into SPPs by gratings and Cosine-Gaussian plasmons can be generated using a periodic arranged array. Compared to Airy plasmons, CGBs with the unique properties could have potential applications in areas such as near-field plasmonic manipulation of particles [32], nano-photonic devices [33] and the transmission of plasmonic signals among on-chip devices [34].

3. Result and discussion

In this paper, we let the relaxation time of graphene $τ = 0.5 ps$ , which is determined by the carrier mobility $μ$ in graphene. The choice of $τ = 0.5 ps$ here is rather conservation to reflect the practical transport loss of graphene [35]. The temperature is $T = 300 K$ . At the same time, the frequency $f$ is varied between 9THz and 10THz and the chemical potential $μ_{c}$ is varied between 0.1 eV and 0.4 eV, which fulfills the approximation condition $k_{B} T < < | μ_{c} |, ℏ ω$ [28]. Meanwhile, it is worth to note that in order to insure the CGBs with nearly non-diffracting behavior, the waist $w_{0}$ should be larger enough than the oscillation period, and the maximum non-diffraction length $w_{0} / \sin θ$ should be larger enough than the length of the period of gratings along the x axis, which can be estimated by $λ_{s p} / \cos θ$ [10]. As the half angle of the two plane waves propagating normal to the grooves gratings, the angle $θ$ can’t be tuned by the chemical potential and should be in the limit of a small angle to insure the surface waves propagating in the x direction without diffraction and enhance the maximum non-diffracting length.

When the relative permittivity of the two dielectrics are assumed as $ε_{1} = 1$ and $ε_{2} = 1.5$ for simplicity. And we fix the frequency $f = 10 THz$ . Figures 2(a) and 2(b) show the dependences between the chemical potential and the real and imaginary parts of the surface conductivity of graphene $σ_{g}$ , respectively. Both the real and imaginary parts increase with the increasing of the chemical potential. According to Eq. (8), both the real and imaginary parts of the wave vector decrease if the chemical potential is tuned from 0.1eV to 0.4eV, as shown in Figs. 2(c) and 2(d). Under a fixed angle, according to Eq. (6) and Eq. (7), the changes of the wave vector $k_{s p}$ have a direct effect on the components of the wave vector. As shown in Figs. 2(e) and 2(f), the real part of $k_{y}$ and the imaginary part of $k_{x}$ decrease with the chemical potential increasing, which lead to a larger transverse oscillation period and a larger propagation length, respectively.

Fig. 2 The dependences between the chemical potential and (a) the real part of the surface conductivity of graphene, (b) the imaginary part of the surface conductivity of graphene, (c) the real part of the wave vector $k_{s p}$ , (d) the imaginary part of the wave vector $k_{s p}$ , (e) the real part of the wave vector $k_{y}$ and (f) the imaginary part of the wave vector $k_{x}$ , respectively. The parameters are $f = 10 THz$ , $τ = 0.5 ps$ , $T = 300 K$ , $θ = 1^{\circ}$ , $ε_{1} = 1$ and $ε_{2} = 1.5$ .

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Figure 3(a) shows the electric field intensity distributions of ${| E_{z} |}^{2}$ at different cut planes, where the parameters are $w_{0} = 200 μm$ , $T = 300 K$ , $f = 10 THz$ , $θ = 1^{\circ}$ and $μ_{c} = 0.4 eV$ . The intensity at y-z plane is governed by the localized TM plasmonic mode of the graphene plasmonic waveguide, as shown in Figs. 3(b) and 3(c). At the x-z plane with y = 0, we can see that the transverse beam profile decays exponentially along the propagation direction, as shown in the Fig. 3(d). Along the interface (z = 0), the field distribution exhibits a straight propagation trajectory of the CGB, as shown in the Fig. 3(e).

Fig. 3 (a) The distributions of for a CGB on three y-z cut planes with $x = 0 μm$ , $x = 5 μm$ and $x = 12 μm$ , respectively, and on x-y cut planes along the propagation direction with z = 0 and on x-z plane with y = 0. The distributions at the y-z plane with $x = 0 μm$ and $x = 12 μm$ are shown in (b) and (c), respectively. The distributions at the x-z plane with y = 0 and the x-y plane with z = 0 are shown in (d) and (e), respectively. The other parameters are $f = 10 THz$ , $τ = 0.5 ps$ , $T = 300 K$ , $θ = 1^{\circ}$ , $w_{0} = 200 μm$ , $μ_{c} = 0.4 eV$ , $ε_{1} = 1$ and $ε_{1} = 1.5$ .

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In order to dynamically manipulate the CGBs, Fig. 4 shows the electric field intensity distributions ${| E_{z} |}^{2}$ at the interface for different chemical potentials, where the $w_{0} = 200 μm$ , $T = 300 K$ , $θ = 1^{\circ}$ and $f = 10 THz$ . As shown in Figs. 4(a)-4(d), we can clearly see the propagation length and oscillation period of the CGBs are growing with the increasing of the chemical potential. When the chemical potential increases, both the real and imaginary parts of the wave vector $k_{s p}$ decrease. Since the angle is fixed, the imaginary part of $k_{x}$ and the real part of $k_{y}$ also decrease. It is precisely the decrease of imaginary part of $k_{x}$ and real part of $k_{y}$ leads to a larger propagation length and a larger oscillation period, respectively. Thus both the propagation length and oscillation period of the CGBs are dynamically manipulated by changing the chemical potential of graphene.

Fig. 4 The electric filed intensity distribution ${| E_{z} |}^{2}$ of a CGB at the x-y plane with z = 0 for different chemical potentials (a) $μ_{c} = 0.1 eV$ , (b) $μ_{c} = 0.2 eV$ , (c) $μ_{c} = 0.3 eV$ and (d) $μ_{c} = 0.4 eV$ . The other parameters are $f = 10 THz$ , $τ = 0.5 ps$ , $T = 300 K$ , $w_{0} = 200 μm$ , $θ = 1^{\circ}$ , $ε_{1} = 1$ and $ε_{2} = 1.5$ .

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The frequency and the angle also affect the dynamical propagation of the CGBs. For quantitative analysis the behavior of the Cosine-Gauss Beams, the analytically estimated propagation length $L_{C G B}$ satisfies $L_{C G B} = 1 / 2 {k^{″}}_{x}$ [8] and the oscillation period $T_{C G B}$ is equal to $2 π / {k^{'}}_{y}$ , where ${k^{″}}_{x} = Im (k_{x})$ and ${k^{'}}_{y} = Re (k_{y})$ . In the case of a fixed chemical potential $μ_{c} = 0.3 eV$ and a fixed angle $θ = 1^{\circ}$ , Figs. 5(a)-5(d) show that when the frequency increase, the real part of the wave vector $k_{y}$ and the imaginary part of the wave vector $k_{x}$ both increase, and the propagation length $L_{C G B}$ and the oscillation period $T_{C G B}$ both decrease. Figures 5(e) and 5(f) show the electric field intensity distributions ${| E_{z} |}^{2}$ at the interface for $f = 9 THz$ and $f = 10 THz$ , respectively. The other parameters are $w_{0} = 200 μm$ and $T = 300 K$ . We can intuitively see the propagation length and oscillation period of the CGBs both decrease with the increasing of the frequency.

Fig. 5 The dependences between the frequency and (a) the real part of the wave vector $k_{y}$ , (b) the imaginary part of the wave vector $k_{x}$ , (c) the propagation length $L_{C G B}$ and (d) the oscillation period $T_{C G B}$ of a CGB, respectively. The electric filed intensity distribution ${| E_{z} |}^{2}$ of a CGB at the x-y plane with z = 0 for different frequency (e) $f = 9 THz$ , (f) $f = 10 THz$ , The parameters are $τ = 0.5 ps$ , $T = 300 K$ , $θ = 1^{\circ}$ , $w_{0} = 200 μm$ , $μ_{c} = 0.3 eV$ , $ε_{1} = 1$ and $ε_{2} = 1.5$ .

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Figure 6 shows the influence of the angle $θ$ to the dynamical propagation of the CGBs in the case of a fixed chemical potential $μ_{c} = 0.3 eV$ . The other parameters are $w_{0} = 200 μm$ , $T = 300 K$ and $f = 10 THz$ . Figures 6(a)-6(d) show that when the angle $θ$ increase, the real part of the wave vector $k_{y}$ increase and the imaginary part of the wave vector $k_{x}$ decrease, and the propagation length $L_{C G B}$ increase and the oscillation period $T_{C G B}$ decrease, respectively. Figures 6(e) and 6(f) show the electric field intensity distributions ${| E_{z} |}^{2}$ at the interface for $θ = 1^{\circ}$ and $θ = 2^{\circ}$ , respectively. We can see intuitively a larger angle leads to a larger propagation length and a smaller oscillation period of CGBs.

Fig. 6 The dependences between the angle $θ$ and (a) the real part of the wave vector $k_{y}$ , (b) the imaginary part of the wave vector $k_{x}$ , (c) the propagation length $L_{C G B}$ and (d) the oscillation period $T_{C G B}$ of a CGB, respectively. The electric filed intensity distribution ${| E_{z} |}^{2}$ of a CGB at the x-y plane with z = 0 for different angle (e) $θ = 1^{\circ}$ and (f) $θ = 2^{\circ}$ . The parameters are $f = 10 THz$ , $τ = 0.5 ps$ , $T = 300 K$ , $w_{0} = 200 μm$ , $μ_{c} = 0.3 eV$ , $ε_{1} = 1$ and $ε_{2} = 1.5$ .

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In order to further study the effect of different material on the dynamical behavior of CGBs, we define the refractive index difference $Δ n = \sqrt{ε_{2}} - \sqrt{ε_{1}}$ , as in general $μ_{1} = μ_{2} \approx 1$ . Here we fix $ε_{1} = 1$ , and let the $ε_{2}$ from 1 to 4, for simplicity. Thus the refractive index difference $Δ n$ will be varied from 0 to 1. Figures 7(a)-7(d) show that when the refractive index difference increase, both the real part and the imaginary part of the wave vector $k_{s p}$ increase, but the propagation length $L_{C G B}$ and the oscillation period $L_{C G B}$ both decrease. The dynamical behavior of CGBs can be seen intuitively, as shown in the Figs. 7(e) and 7(f). Thus the smaller the refractive index difference $Δ n$ , the larger propagation length and the larger oscillation period of CGBs.

Fig. 7 The dependences between the refractive index difference $Δ n$ and (a) the real part of the wave vector $k_{s p}$ , (b) the imaginary part of the wave vector $k_{s p}$ , (c) the propagation length $L_{C G B}$ and (d) the oscillation period $T_{C G B}$ of a CGB, respectively. The electric filed intensity distribution ${| E_{z} |}^{2}$ of a CGB at the x-y plane with z = 0 for different refractive index difference (e) $Δ n = 0$ and (f) $Δ n = 1$ , respectively. The parameters are $f = 10 THz$ , $τ = 0.5 ps$ , $T = 300 K$ , $w_{0} = 200 μm$ , $θ = 1^{\circ}$ , $μ_{c} = 0.3 eV$ and $ε_{1} = 1$ .

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4. Conclusion

In conclusion, we show that the CGBs are supported on a graphene plasmonic waveguide. Both the transverse oscillation period and propagation length of CGBs can be dynamically manipulated by changing the chemical potential of graphene. The graphene-based planar plasmonic waveguide provides a platform with dynamical manipulation to investigate the propagation dynamics of CGBs, as well as other non-diffracting beam. Besides, the decrease of the different refractive index difference of the graphene plasmonic waveguide can lead to a larger propagation length and oscillation period of CGBs. Those interesting finding may lead to the flexible optical micromanipulation in flatland plasmonic devices and chip scale signal processing. The development of some special materials that can support low-loss plasmons may shed new light on the propagation of CGBs in graphene-based planar plasmonic waveguide, such as graphene-boron nitride heterostructures [36, 37], multilayer graphene [38, 39] and gain material [40], etc. Our work provides a theoretical guidance to the further structure design and optimization in plasmonic signal processing.

Funding

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 61471033 and 61525501.

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Dynamical manipulation of Cosine-Gauss beams in a graphene plasmonic waveguide

Abstract

1. Introduction

2. Structure and model

3. Result and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (9)

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