Tunable Goos-H&#x000E4;nchen shift from graphene ribbon array

Xiaodong Zeng; M. Al-Amri; M. Suhail Zubairy

doi:10.1364/OE.25.023579

1. Introduction

Goos-Hänchen (GH) shift refers to a lateral shift between the reflected and the incident beams, which is named after Goos and Hänchen who observed this effect firstly by using multiple reflections in a glass slab [1–15]. Afterwards, Artmann and other authors theoretically interpreted this effect [2]. In their theories, the stationary light beam is expanded as a power series of plan-wave components. After the reflection, different components undergo different phase and amplitude changes, which results in a total spatial lateral shift. Lots of structures for generating large GH shifts have been investigated [4–12], such as waveguide, photonic crystal, and left-handed metamaterials, including both positive and negative shifts. Additionally, obvious GH shift also occurs in a surface grating [7–12].

Graphene, a single layer of carbon atoms arranged in a honeycomb lattice, has attracted tremendous interest [16–36]. The linear dispersion relation near the Dirac point of the energy band induces a special optical response to the light. Especially the graphene plasmons (GPs) have emerged as a hot topic in recent years due to their new frequency region, tunability, long-lived and extreme light confinement. Additionally, due to Pauli-blocking effect, doped graphene has a low absorption in the terahertz to mid-infrared.

To investigate the reflectivity of light beam by graphene, we need to enhance the coupling between the beam and the graphene due to little light can be reflected by monolayer graphene at small incident angle. One practical way is to excite the GPs [21]. However, as we know, GPs have large wave numbers [22] and are hard to be excited by a beam incident on the graphene directly. Periodic array of graphene ribbons, which can be considered to be a grating [37], has attracted a lot of attention because the GPs can be easily excited in the system [30, 34]. Light beam incident on the array can be converted to the leaky resonant graphene plasmons (RGPs) efficiently if the ribbon width satisfies the resonance conditions, which will induce large reflection. Additionally, since light waves with different incident angles have different coupling strengths as well as phase changes after the reflection, we can expect large GH shifts of the reflection fields.

In this article, formulas to calculate the reflections as well as GH shifts of light beams by Green’s tensor method are derived. The numerical results indicates that light beams incident on graphene ribbon array have either large positive or negative GH shifts at certain incident angles, which is qute different with monolayer graphene [38]. And, more remarkable, due to the tunability of graphene Fermi level, the plasmonic resonance conditions can be manipulated and consequently the GH shift can be controlled conveniently. These effects are helpful to the investigation of graphene-based metasurface [30, 39].

The paper is organized as follows. In Sec. 2, we introduce our model and mathematic theory. In Sec. 3, we present some numerical results and simulations. In Sec. 4, we present the concluding remarks.

2. Green’s tensor method to solve the scattering

Our model is schematically shown in Fig. 1. The graphene ribbon array is located at z = 0 along y direction and between two dielectric half spaces with dielectric constants ε₁ and ε₂, respectively. The ribbon has a width b and period a. A Gaussian beam is incident on the array with p − z plane as the incident plane. The beam direction has a polar angle θ (with respect to the z axis) and an azimuthal angle φ (with respect to the x axis).

Fig. 1 A Gaussian beam is incident on the periodic graphene ribbon array. The Fermi level of the ribbons can be tuned by the gate voltage.

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In the long wavelength and high doping limit, i.e., ħω ⩾ 2E_F, the in-plane conductivity of graphene can be described as [25, 28]

σ (ω) = \frac{i e^{2} E_{F}}{π ℏ^{2} (ω + i τ^{- 1})}

under the random phase approximation (RPA). Here, E_F is the Fermi level, e is the electron charge, and τ describes the momentum relaxation time. Under certain conditions, τ is identified with DC relaxation time and can be expressed as

τ = μ E_{F} / e v_{F}^{2}

[33], where μ is the DC mobility and can reach 10⁴⁻⁶cm²/Vs [29]. The tunability of the graphene originates from the controllability of the Fermi level. For example as in Fig. 1, the Fermi level can be manipulated by controlling the gate voltage.

Here, we use Green’s tensor method to deal with the scattering of the light. A plane wave with an electric field component parallel to the x − y plane is incident on the ribbon array can induce surface electric current η(x, y) in the graphene layer. The field produced by each surface element dx′ dy′ is E(x, y) = G(x − x′, y − y′; ω)·(iη(x′, y′)/ω)dx′ dy′, where G is the Green’s tensor [25, 40, 41]. Summing over all of these surface current contributions, and including the effect of a substrate through its Fresnel coefficients, we obtain the self-consistent relations (see the Appendix)

\begin{array}{l} \frac{η_{x} (x, y)}{σ (x, y)} = E_{x}^{0} (x, y, 0) + \frac{1}{2 {(2 π)}^{3}} \frac{i}{ω} \int \int \int \int d k_{x} d k_{y} d x^{'} d y^{'} \\ \frac{2 π i}{β} \frac{k_{0}^{2}}{ε_{1}} {[\frac{k_{y}^{2}}{ρ^{2}} (1 + r_{s}) + \frac{1}{k_{0}^{2}} \frac{k_{x}^{2} β^{2}}{ρ^{2}} (1 - r_{p})] η_{x} (x^{'}, y^{'}) + \\ [\frac{- k_{x} k_{y}}{ρ^{2}} (1 + r_{s}) + \frac{1}{k_{0}^{2}} \frac{k_{x} k_{y} β^{2}}{ρ^{2}} (1 - r_{p})] η_{y} (x^{'}, y^{'})} e^{i [k_{x} (x - x^{'}) + k_{y} (y - y^{'})]}; \end{array}

and

\begin{array}{l} \frac{η_{y} (x, y)}{σ (x, y)} = E_{y}^{0} (x, y, 0) + \frac{1}{2 {(2 π)}^{3}} \frac{i}{ω} \int \int \int \int d k_{x} d k_{y} d x^{'} d y^{'} \\ \frac{2 π i}{β} \frac{k_{0}^{2}}{ε_{1}} {[\frac{- k_{x} k_{y}}{ρ^{2}} (1 + r_{s}) + \frac{1}{k_{0}^{2}} \frac{k_{x} k_{y} β^{2}}{ρ^{2}} (1 - r_{p})] η_{x} (x^{'}, y^{'}) \\ + [\frac{- k_{x}^{2}}{ρ^{2}} (1 + r_{s}) + \frac{1}{k_{0}^{2}} \frac{k_{y}^{2} β^{2}}{ρ^{2}} (1 - r_{p})] η_{y} (x^{'}, y^{'})} e^{i [k_{x} (x - x^{'}) + k_{y} (y - y^{'})]}, \end{array}

where η_x and η_y are the induced surface electric currents along x and y directions,

ρ = {(k_{x}^{2} + k_{y}^{2})}^{1 / 2}

and

β = {(ε_{1} k_{0}^{2} - ρ^{2})}^{1 / 2}

is the wave vector perpendicular to the ribbon array. r_s and r_p are the Fresnel reflection coefficients of the substrate for TE and TM polarized waves.

E_{x}^{0}

and

E_{y}^{0}

are the external electric fields components which already include the reflection by the homogeneous dielectric substrate.

Since the conductivity is periodic along the x direction and homogenous along the y direction, we can expand the conductivity along the x direction as Fourier series given by

σ (x, y) = \sum_{n} σ_{n} e^{i g_{n} x},

with g_n = nΛ, where Λ = 2π/a is the reciprocal wave vector of the ribbon array. The surface current can be expanded as

η_{x, y} (x, y) = \sum_{n} η_{x, y}^{n} e^{i [(k_{x 0} + g_{n}) x + k_{y 0} y]},

where

k_{x}_{0} = \sqrt{ε_{1}} k_{0} \cos θ

and

k_{y}_{0} = \sqrt{ε_{1}} k_{0} \sin θ

sin θ are the wave number components of the incident field along x and y directions, respectively. After some straightforward algebra, we project Eqs. (2) and (3) into

\begin{array}{l} η_{x}^{n} = \frac{1}{a} \int_{0}^{b} d x E_{x}^{0} (x, y, 0) σ e^{- i (k_{x} x + k_{y} y)} - \frac{1}{2 ω} \sum_{n^{'}} \frac{k_{0}^{2}}{ε_{1} β} {[\frac{k_{y}^{2}}{ρ^{2}} (1 + r_{s}) + \\ \frac{1}{k_{0}^{2}} \frac{k_{x}^{2} β^{2}}{ρ^{2}} (1 - r_{p})] η_{x}^{n^{'}} σ_{n - n^{'}} + [\frac{- k_{x} k_{y}}{ρ^{2}} (1 + r_{s}) + \frac{1}{k_{0}^{2}} \frac{k_{x} k_{y} β^{2}}{ρ^{2}} (1 - r_{p})] η_{y}^{n^{'}} σ_{n - n^{'}}}; \end{array}

and

\begin{array}{l} η_{y}^{n} = \frac{1}{a} \int_{0}^{b} d x E_{y}^{0} (x, y, 0) σ e^{- i (k_{x} x + k_{y} y)} - \frac{1}{2 ω} \sum_{n^{'}} \frac{k_{0}^{2}}{ε_{1} β} {[\frac{- k_{x} k_{y}}{ρ^{2}} (1 + r_{s}) + \\ \frac{1}{k_{0}^{2}} \frac{k_{x} k_{y} β^{2}}{ρ^{2}} (1 - r_{p})] η_{x}^{n^{'}} σ_{n - n^{'}} + [\frac{k_{x}^{2}}{ρ^{2}} (1 + r_{s}) + \frac{1}{k_{0}^{2}} \frac{k_{y}^{2} β^{2}}{ρ^{2}} (1 - r_{p})] η_{y}^{n^{'}} σ_{n - n^{'}}} . \end{array}

Here, β and ρ are the same as those in Eqs. (2) and (3) but for k_x = k_x₀ + g_n_′ and k_y = k_y₀. Finally, we solve Eqs. (4) and (5) by using standard linear algebra with a finite number of waves M. The zeroth-order reflection field is given in terms of the η⁰ coefficients as

\begin{array}{l} E_{x} (x, y, z) = E_{x}^{r} (x, y, z) - \frac{i}{2 ω} \frac{k_{0}^{2}}{ε_{1} β} e^{i [k_{x 0} x + k_{y 0} y + β_{z}]} {[\frac{k_{y 0}^{2}}{ρ^{2}} (1 + r_{s}) + \frac{1}{k_{0}^{2}} \frac{k_{x 0}^{2} β^{2}}{ρ^{2}} (1 - r_{p})] η_{x}^{0} + \\ [\frac{- k_{x 0} k_{y 0}}{ρ^{2}} (1 + r_{s}) + \frac{1}{k_{0}^{2}} \frac{k_{x 0} k_{y 0} β^{2}}{ρ^{2}} (1 - r_{p})] η_{y}^{0}} = | r_{x} | e^{i ϕ} E_{x}^{0} (x, y, z) e^{2 i β z}; \end{array}

and

\begin{array}{l} E_{y} (x, y, z) = E_{y}^{r} (x, y, z) - \frac{i}{2 ω} \frac{k_{0}^{2}}{ε_{1} β} e^{i [k_{x 0} x + k_{y 0} y + β_{z}]} {[\frac{- k_{x 0} k_{y 0}}{ρ^{2}} (1 + r_{s}) + \frac{1}{k_{0}^{2}} \frac{k_{x 0} k_{y 0} β^{2}}{ρ^{2}} (1 - r_{p})] η_{x}^{0} \\ + [\frac{k_{x 0}^{2}}{ρ^{2}} (1 + r_{s}) + \frac{1}{k_{0}^{2}} \frac{k_{y 0}^{2} β^{2}}{ρ^{2}} (1 - r_{p})] η_{y}^{0}} = | r_{y} | e^{i ϕ} E_{y}^{0} (x, y, z) e^{2 i β z} . \end{array}

Here,

E_{x}^{r} (x, y, z)

and

E_{y}^{r} (x, y, z)

are the direct reflection of the incident field by the dielectric substrate, and ϕ is the phase shift.

For the incident beam with a sufficiently large beam waist (i.e., the beam with a narrow angular spectrum, Δk ⩽ k₀, where Δk = 1/w₀ and w₀ is the half width of the beam at waist), the GH shift of the zeroth-order reflection beam can be expressed as [4]

S = - \frac{λ_{0}}{2 π} \frac{d ϕ}{d θ},

where λ₀ is the vacuum wavelength.

3. Numerical calculations of Goos-Hänchen shift

In this section, we numerically calculate the reflectivities, phase changes, and GH shifts of the reflection fields. In the calculations, we set M to be 700, which can give us precious numerical results. We focus on two cases that TM modes with azimuthal angle ϕ to be 0 and TE mode with ϕ to be π/2.

In Fig. 2, we show the zeroth-order reflection spectrums for light waves incident on the array perpendicularly. Here, b = 4.8μm, a = 49.8μm, and τ = 1ps. For the convenience of calculations, we set ε₁ = ε₂ = 1. Since a ⩾ b, we can neglect the interference between the ribbons. Remarkably, the ribbon is wide enough that we don’t need to consider the edge effect of the ribbon and adopt the conductivity shown in Eq. (1) [36]. The peak frequencies satisfy the plasmonic resonance conditions

\frac{n π}{b} \approx \frac{(1 + ε_{2}) ω^{2}}{4 α_{0} c ω_{F}} .

Here ω_F = E_F/ħ, α₀ is the fine-structure constant, and integers n = 1, 2, 3… measure the number of half wavelengths that fit within the ribbon width. The right three high peak frequencies correspond to n = 1 (electric dipole resonance), denoted by

ω_{E_{F}}^{1}

, and the three left low peak frequencies correspond to n = 2 (electric quadrupole resonance), denoted by

ω_{E_{F}}^{2}

[34]. The high reflectivity originates from the strong coupling strength between the incident field and the RGPs. When E_F = 0.6eV, the vacuum wave number of the peak frequency

ω_{0.6}^{1}

is larger than Λ, thus the first-order diffraction field is still in the far field region and has weak coupling with the RGPs. The reflection mainly comes from the coupling between the higher-order diffraction modes and the RGPs. However, when E_F = 0.4, 0.5eV, the first-order diffraction fields with frequencies

ω_{0.4, 0.5}^{1}

are evanescent waves and concentrated in the proximity of the ribbon array. This leads to stronger coupling with the RGPs and consequently higher reflectivities.

Fig. 2 The zeroth-order reflection spectrum for the ribbon array with different Fermi levels.

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In Fig. 3(a), we show the reflection of TM waves incident on the array with an incident angle θ. Two peaks appear in the spectrum on the Rayleigh anomalies [42]. The left (right) peak position corresponds to θ₁ = arcsin[(k₀ − Λ)/k₀] (θ₋₂ = arcsin[(2Λ − k₀)/k₀]), where the plus first (minus second)-order diffraction component of the incident wave becomes evanescent wave and has large coupling strength with the RGPs. As θ increases, the x-direction electric field of the incident beam decreases, which leads to smaller surface currents [25] and consequently decreasing reflectivity as shown in Fig. 3(a).

Fig. 3 (a) The zeroth-order reflectivities of TM waves with φ = 0 at different Fermi levels. The labels (1) and (2) represent the plus first and minus second-order diffraction positions. The wavelength of the incident beam is 45.2μm and the other parameters are the same as Fig. 2. (b) The arguments ϕ of the zeroth-order reflection field. (c) and (d) The corresponding GH shifts.

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The GH shift can be calculated by using Eqs. (8–10). If ε₂ = 1 and we just consider the zeroth-order reflection, $η_{x}^{0}$ dominates the reflection field that we are interested in, thus we can obtain the phase shift

ϕ = A r g (η_{x}^{0}) .

In Fig. 3(b), we plot ϕ. When E_F = 0.6eV, intense variations and even sudden jumps happen, especially near the Rayleigh anomalies. The corresponding GH shifts are shown in Figs. 3(c) and (d). We can see giant GH shifts when E_F = 0.6eV. These results can be understood by Artmann’s theory. As the incident angular approaches and exceeds θ₁, the z direction wave vector of the first-order diffraction field varies from real to zero and then to pure imaginary. This means that the coupling strength between the diffraction field and the RGPs increases intensely and the phase ϕ varies sharply. As a consequence, large GH shift appears. However, when the incident angular is much larger than θ₁, the coupling strength and phase ϕ vary relatively slowly, therefore the total GH shift approaches zero [4]. It can be interpreted similarly for the case that the incident angle approaches θ₋₂. Additionally, as we know, the derivation of Eq. (10) in the article is based on the condition that the phase gradient is almost a constant in the angular spectrum. This means that at the points θ = θ₁ and θ₋₂, Eq. (10) is not valid anymore. This is why there are poles in Figs. 3(c) and (d). However, a real optical beam can be expended as a series of plane waves around these angles and the total GH shift is the coherent summation of the contributions from all the plane waves. The waves with the incident angles θ₁ and θ₋₂ will have little influence on the total GH shifts and the GH shifts will always be finite. In the following, we will give some simulations for real optical beams.

When E_F = 0.4 or 0.5eV, the incident fields are not resonant with the ribbon array and there are no RGPs. The coupling strengths between the incident beams and array are weak and the reflectivities are small as shown in Fig. 3(a). Meanwhile, since all the components of the incident beams have small coupling strengths with the ribbons, which are similar to the cases that the beams are reflected by low reflection dielectric interfaces, the beams have small GH shifts.

Additionally, the Fermi level of the ribbon can be tuned easily, such as by static electric gate shown in Fig. 1. This gives us a way to manipulate the GH shift. In Fig. 4, we plot the GH shifts as a function of the Fermi level. The curves shows that the GH shifts are highly dependent on the Fermi level. Compared to a small ε₂, the resonance condition of Eq. (11) requires large E_F for a large ε₂. We also plot the GH shifts for the ribbons with different relaxation time. For the electric dipole resonance, the polarizability is inverse proportion to the dissipation [34]. As a consequence, large dissipation weakens the coupling between the incident field and the RGPs and slows the φ variation. The GH shift decreases with the dissipation as shown in Fig. 4.

Fig. 4 (The GH shifts versus the Fermi level under different ε₂ and τ. The parameters are the same as the Fig. 3(a) and θ = 0.0769.

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The reflection spectrums and GH shifts for the TE waves with φ = π/2, i.e., the waves are incident in the y − z plane, are shown in Fig. 5. In contrast to Fig. 3(a) where there are two peaks, there is only one valley in the spectrum at position $θ = \arcsin \sqrt{1 - Λ^{2} / k_{0}^{2}}$ . The reflectivities approach zero near this point. Meanwhile, both positive and negative GH shifts occur when E_F = 0.6eV and only positive shifts exist when E_F = 0.4 and 0.5eV. Similar to the case that we discussed in Fig. 4, we can also manipulate the reflectivity and GH shift by controlling the Fermi level of the graphene ribbons.

Fig. 5 (a) The zeroth-order reflectivity of TE wave along y direction asa function the incidence angle at different Fermi levels. (b) The corresponding GH shifts.

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In order to verify our previous theoretical predictions, we simulate the reflection fields of TM polarized Gaussian beams in Fig. 6. The vector electric field of the beam is expressed in terms of its angular spectrum as follows [3, 43]:

E (r) = \int \int A (k_{x 0}, k_{y 0}) e^{i k \cdot r} d k_{x 0} d k_{0},

where time dependence exp(−iωt) is assumed and suppressed. This beam has a principal axial direction k^G = (k₀ sin θ₀, 0, k₀ cos θ₀). k = (k_x₀, k_y₀, β₀) is the wave vector satisfying

k_{x 0}^{2} + k_{y 0}^{2} + β_{0}^{2} = k_{0}^{2}

. The element A(k_x₀, k_y₀) with direction vector (cos θ cos φ, − cos θ sin φ, sin θ) has the following amplitude [3, 43]

A (k_{x 0}, k_{y 0}) = {(\frac{w_{x} w_{y}}{π})}^{1 / 2} \exp [- \frac{w_{x}^{2}}{2} {(k_{x 0} -_{x 0}^{G})}^{2} - \frac{w_{y}^{2}}{2} k_{y 0}^{2}],

where w_x = w₀/cos θ₀ and w_y = w₀. In our simulation,

w_{0} = 250 \sqrt{2} λ_{0},

θ₀ = 0.0771 and the other parameters are the same as those in Fig. 3. We utilize the method in the previous section to calculate the reflection fields for all the plane waves k and then summarize these fields associated with the element amplitudes A(k_x₀, k_y₀) to obtain the beam intensity distribution.

Fig. 6 (a) The normalized field intensity distribution of the incident TM-polarized Gaussian beam on the z = 0 plane. (b, c) The normalized field distributions of the zeroth-order reflection fields for the cases that E_F = 0.6eV and 0.5eV.

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In Fig. 6(b), we show that when E_F = 0.6eV, the zeroth-order reflection beam has a lateral shift 3.14λ₀, which is approximately equal to the previous theoretical result, i.e., 3.02λ₀/cos θ₀ = 3.27λ₀. While for E_F = 0.5eV, the reflection beam nearly has the same position as the incident beam, which also agrees with the previous prediction. These results evidently prove that the GH shift from graphene ribbon array is highly dependent on the Fermi level.

4. Conclusion

In conclusion, we calculate the reflections and GH shifts from graphene ribbon array by Green’s tensor method. Both positive and negative GH shifts can be found. Compared to the monolayer graphene where prism is required to excite the GPs and consequently GH shift [44–46], graphene ribbon array is a simpler and more practical choice to realize giant GH shifts. Additionally, it is easy to control the graphene Fermi level [30], meanwhile, the GH shift is strongly dependent on the Fermi level. Therefore, we can control the GH shifts conveniently. Our investigation is important to the tunable metasurfaces and electro-optical devices based on graphene. Compared with the monolayer graphene used in Refs. 45 and 46, realizing the periodical ribbon array is technologically harder. However, periodical ribbon array systems are widely investigated in other topics and the technology is not a big issue now, such as in Ref. 30 in the revised manuscript. We hope and also believe the experiment can be realized in the near future

Appendix

In this Appendix, we present the derivations of equations (2) and (3). According to the Maxwell’s equation, the electric field E(r, t) = E(r, ω)exp(−iωt) of the oscillatory electric current η(r, t) = η(r, ω)exp(−iωt) satisfies the equation [41]

(\nabla \times \nabla \times - ε (r, ω) \frac{ω^{2}}{c^{2}}) E (r, ω) = \frac{i}{ω} η (r, ω) .

In the absence of an external field, E(r, ω) is formally given by

E (r, ω) = \frac{i}{ω} \int d^{3} r^{'} G (r, r^{'}; ω) \cdot η (r^{'}, ω) .

In our system, the Green’s tensor can be described as the summation of plane waves

\begin{array}{l} G_{x x} (r_{∥}, r_{∥}^{'}; ω) = \frac{1}{2 {(2 π)}^{3}} \int \int d k_{x} d k_{y} \frac{2 π i}{ε_{1} β} [\frac{k_{y}^{2}}{ρ^{2}} (1 + r_{s}) \\ + \frac{k_{x}^{2} β^{2}}{k_{0}^{2} ρ^{2}} (1 - r_{p})] e^{i [k_{x} (x - x^{'}) + k_{y} (y - y^{'})]}; \end{array}

\begin{array}{l} G_{x y} (r_{∥}, r_{∥}^{'}; ω) = \frac{1}{2 {(2 π)}^{3}} \int \int d k_{x} d k_{y} \frac{2 π i}{ε_{1} β} [\frac{- k_{x} k_{y}}{ρ^{2}} (1 + r_{s}) \\ + \frac{k_{x} k_{y} β^{2}}{k_{0}^{2} ρ^{2}} (1 - r_{p})] e^{i [k_{x} (x - x^{'}) + k_{y} (y - y^{'})]}; \end{array}

G_{y x} (r_{∥}, r_{∥}^{'}; ω) = G_{x y} (r_{∥}, r_{∥}^{'}; ω);

\begin{array}{l} G_{y y} (r_{∥}, r_{∥}^{'}; ω) = \frac{1}{2 {(2 π)}^{3}} \int \int d k_{x} d k_{y} \frac{2 π i}{ε_{1} β} [\frac{k_{x}^{2}}{ρ^{2}} (1 + r_{s}) \\ + \frac{k_{y}^{2} β^{2}}{k_{0}^{2} ρ^{2}} (1 - r_{p})] e^{i [k_{x} (x - x^{'}) + k_{y} (y - y^{'})]} . \end{array}

Here r_|| = (x, y, 0) denotes the spot on the z = 0 plane. The reflection coefficients

r_{s} = (β - \sqrt{ε_{2} k_{0}^{2} - ρ^{2}}) / (β + \sqrt{ε_{2} k_{0}^{2} - ρ^{2}});

and

r_{p} = (ε_{2} β - ε_{1} \sqrt{ε_{2} k_{0}^{2} - ρ^{2}}) / (ε_{2} β + ε_{1} \sqrt{ε_{2} k_{0}^{2} - ρ^{2}}) .

Eqs. (16–19) can be understood as the x(y) direction field at position r_║ induced by a unit x(y) direction current at position $r_{∥}^{'}$ . By using relation η = σE, we can obtain equations (2) and (3).

Funding

Qatar National Research Fund (QNRF) NPRP Grant No. 8-352-1-074; King Abdulaziz City for Science and Technology (KACST).

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Tunable Goos-Hänchen shift from graphene ribbon array

Abstract

1. Introduction

2. Green’s tensor method to solve the scattering

3. Numerical calculations of Goos-Hänchen shift

4. Conclusion

Appendix

Funding

References and links

Cited By

Figures (6)

Equations (22)

Optics Express