Goos-H&#x000E4;nchen and Imbert-Fedorov shifts for Gaussian beams impinging on graphene-coated surfaces

Simon Grosche; Marco Ornigotti; Alexander Szameit

doi:10.1364/OE.23.030195

1. Introduction

When an optical beam impinges upon a surface, nonspecular reflection phenomena may occur, such as the Goos-Hänchen (GH) [1–3] and Imbert-Fedorov (IF) [4, 5] shifts, resulting in an effective beam shift at the interface. A comprehensive review on beam shift phenomena can be found in Ref. [6]. Although Goos and Hänchen published their work more than 60 years ago [1], this field of research is still very active, and in the last decades a vast amount of literature has been produced on the subject, resulting not only in a better understanding of the underlying physical principles [7–14], but also in the careful investigation of the effects of various field configurations [15–19] and reflecting surfaces [20–23] on the GH and IF shifts. In the last decade, moreover, the giant GH shift has also been observed to occur in various systems, such as metamaterials [24], photonic crystals [25] and complex crystals [26].

In recent years, on a parallel trail, graphene attracted very rapidly a lot of interest, thanks to its intriguing properties [27, 28]. Its peculiar band structure and the existence of the so-called Dirac cones [29], for example, gives the possibility to use graphene as a model to observe QED-like effects such as Klein tunneling [30], Zitterbewegung [31], the anomalous quantum Hall effect [32] and the appearance of a minimal conductivity that approaches the quantum limit e²/ħ for vanishing charge density [33]. In addition, the reflectance and transmittance of graphene are determined by the fine structure constant [34], and a single layer of graphene shows universal absorbance in the spectral range from near-infrared to the visible part of the spectrum [35].

Among the vast plethora of applications, graphene also proved to be a very interesting system where to observe beam shifts. Very recently, in fact, the occurrence of GH shift in graphene-based structures has been reported, both for light beams (where giant GH shift has been observed [36]) and for Dirac fermions [37]. Contextually, the occurrence of giant GH shift for a graphene layer embedded in two media with different permittivity has been predicted by Martinez and co-workers in 2001 [38]. Despite this theoretical work, however, a full theoretical analysis of both spatial and angular GH and IF shifts has not been carried out yet.

In this work, we therefore present a theoretical analysis of the GH and IF shifts occurring for a monochromatic Gaussian beam impinging onto a glass surface coated with a single layer of graphene. The results of our investigations show on one hand, that the appearance of a giant GH shift is ultimately due to the graphene’s surface conductivity σ(ω) (in accordance with the results of Ref. [38]), and on the other hand, that the presence of the single layer of graphene introduces a dependence of the phases of the reflection coefficients on the incidence angle, thus resulting in a nonzero spatial GH shift also when total internal reflection does not occur. Our results are in agreement with previous investigations of beam shifts in lossy surfaces [39] or multilayered media [17], where the appearance of a nonzero spatial shift due to the complexvaluedness of the reflection coefficient resulting from the nontrivial structure of the surface has also been reported.

We start our analysis by considering a monochromatic Gaussian beam with frequency ω = ck₀ (with k₀ being the vacuum wave number), impinging on a dielectric surface characterized by the refractive index n and coated with a single layer of graphene [Fig. 1(a)]. The graphene layer is characterized by the optical conductivity σ(ω), whose expression can be given in the following dimensionless form [29]

\frac{σ (Ω)}{σ_{0}} = Θ (Ω - 2) + i [\frac{1}{π Ω} - \frac{1}{π} \ln | \frac{Ω + 2}{Ω - 2} |],

where σ₀ = e²/(4ħ) = παcε₀ (being α ≈ 1/137 the fine structure constant [31]) is the universal optical conductivity of graphene, Ω = ħω/µ is the dimensionless frequency, µ is the chemical potential and Θ(x) is the Heaviside step function [40]. Later, we use Ω = 2.5 as an exemplary value for all figures.

Fig. 1 (a) Schematic representation of the considered surface. The graphene layer (green) is characterized by its surface conductivity σ(Ω), whose explicit expression is given by Eq. (1). The dielectric substrate (green) is characterized by the refractive index n. (b) Geometry of beam reflection at the interface. The single graphene layer is located on the surface at z = 0. The different Cartesian coordinate systems K,K_i,K_r are shown.

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According to Fig. 1(b), we define three Cartesian reference frames: the laboratory frame K = (O,x,y,z) attached to the reflecting surface, the (local) incident frame K_i = (O,x_i,y,z_i) attached to the incident beam, and the (local) reflected frame K_r = (O,x_r,y,z_r) attached to the reflected beam. These three reference frames are connected via a rotation of an angle θ around the y direction [39]. The reflecting surface is located at z = 0, with the z-axis pointing towards the interface. With this choice of geometry, the incident beam comes from the region z < 0 and propagates in the x-z plane.

The electric field in the incident frame can be then written, using its angular spectrum representation [41], as follows:

E_{i} (r) = \sum_{^{l = 1}}^{2} \int d^{2} K {\hat{e}}_{λ} (U, V, θ) A_{λ} (U, V, θ) e^{i k_{i} \cdot r_{i}},

where d²K = dUdV, ê_λ (U,V,θ) is the local reference frame attached to the incident field [42], A_λ (U,V,θ) = α_λ (U,V,θ)A(U,V) and k_i · r_i = UX_i +VY_i +WZ_i, being X_i = k₀x_i the normalized coordinate in the incident frame. Y_i and Z_i are defined in a similar manner.

α_{λ} (U, V, θ) = {\hat{e}}_{λ} (U, V, θ) \cdot \hat{f}

accounts for the projection of the beam’s polarization

\hat{f} = f_{p} \hat{x} + f_{s} \hat{y}

(normalized according to |f_p|²+|f_s|²=1) onto the local basis, and A(U,V) is the beam’s spectral amplitude, which here is assumed to be Gaussian, i.e.,

A (U, V) = e^{- w_{0}^{2} (U^{2} + V^{2})},

being

w_{0}^{2}

the spot size of the beam. In the remaining of the manuscript, we will consider only well collimated beams, namely the paraxial assumption U,V ≪ 1 is implicitly understood.

Upon reflection, the electric field can be then written as follows:

E_{i} (r_{r}) = \sum_{^{λ = 1}}^{2} \int d^{2} K {\hat{e}}_{λ} (- U, V, π - θ) {\tilde{A}}_{λ} (U, V, θ) e^{i k_{r} \cdot r_{r}},

where k_r · r_r = −UX_r +VY_r +WZ_r and Ã_λ (U,V,θ) = r_λ (U,V,θ)A_λ (U,V,θ), with r_λ (U,V,θ) being the Fresnel reflection coefficients associated to the single plane wave component of the field [43]. The minus sign in front of U in ê_λ, as well as in k_r · r_r accounts for the specular reflection of the single plane wave component [42].

The presence of a single layer of graphene deposited on the dielectric surface modifies its reflection coefficients as follows [44]:

r_{s} (θ) = \frac{\cos θ - \sqrt{n^{2} - \sin^{2} θ} - σ (Ω)}{\cos θ + \sqrt{n^{2} - \sin^{2} θ} + σ (Ω)},

r_{p} (θ) = \frac{n^{2} \cos θ - \sqrt{n^{2} - \sin^{2} θ} [1 - σ (Ω) \cos θ]}{n^{2} \cos θ + \sqrt{n^{2} - \sin^{2} θ} [1 + σ (Ω) \cos θ]},

where θ is the incident angle, n is the refractive index of the dielectric medium and σ(Ω) is the graphene’s surface conductivity, as defined by Eq. (1). The modulus R_λ and phase ϕ_λ of the reflection coefficients

r_{λ} = R_{λ} e^{i ϕ_{λ}}

(with λ ∈ {p,s}) are shown in Fig. 2, together with the correspondent quantities for the case of a simple dielectric surface without the graphene coating. While the presence of the graphene layer does not modify significatively R_λ for neither p- or s-polarization (as it appears clear from Figs. 2(a) and (c), respectively), the change induced in the phases ϕ_λ of the reflection coefficients is considerable. For an air-glass interface, in fact, we have ∂ϕ_λ/∂θ = 0 being θ the angle of incidence. Here, instead, we have ∂ϕ_λ/∂θ ≠ 0. A closer inspection of Eqs. (5), moreover, reveals that such a novel θ-dependence of the phases ϕ_λ is entirely due to the graphene conductivity σ(Ω).

Fig. 2 Modulus (left column) and phase (right column) of the reflection coefficients r_λ = R_λ exp(iϕ_λ) for p-polarization (top row) and s-polarization (bottom row). In all graphs, the solid black line corresponds to the case of the graphene-coated surface, while the green dashed curve corresponds to the case without graphene coating. The refractive index of the bulk medium is chosen to be n = 1.5.

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To compute the GH and IF shifts, we calculate the center of mass of the intensity distribution in the reflected frame, namely [42]

〈 R 〉 = \frac{{\int \int}_{- \infty}^{+ \infty} R {| E_{r} |}^{2} d X_{r} d Y_{r}}{{\int \int}_{- \infty}^{+ \infty} {| E_{r} |}^{2} d X_{r} d Y_{r}} = 〈 X_{r} 〉 {\hat{X}}_{r} + 〈 Y_{r} 〉 {\hat{y}}_{r},

where R = (X_r,Y_r)^T. Spatial (Δ) and angular (Θ) GH and IF shifts are then defined as follows:

\begin{matrix} k_{0} Δ_{GH} = 〈 X_{r} 〉 |_{z = 0}, & Θ_{GH} = \frac{\partial 〈 X_{r} 〉}{\partial z} \end{matrix},

\begin{matrix} k_{0} Δ_{IF} = 〈 Y_{r} 〉 |_{z = 0}, & Θ_{IF} = \frac{\partial 〈 Y_{r} 〉}{\partial z} \end{matrix} .

The explicit expressions of the GH and IF shifts for a fundamental Gaussian beam read, according to [45], as follows:

k_{0} Δ_{GH} = w_{p} \frac{\partial ϕ_{p}}{\partial θ} + w_{s} \frac{\partial ϕ_{s}}{\partial θ},

k_{0} Δ_{IF} = - \cot θ [\frac{w_{p} a_{s}^{2} + w_{s} a_{p}^{2}}{a_{p} a_{s}} \sin η + 2 \sqrt{w_{p} w_{s}} \sin (η - ϕ_{p} + ϕ_{s})],

Θ_{GH} = - (w_{p} \frac{\partial \ln R_{p}}{\partial θ} + w_{s} \frac{\partial \ln R_{s}}{\partial θ}),

Θ_{IF} = \frac{w_{p} a_{s}^{2} - w_{s} a_{p}^{1}}{a_{p} a_{s}} \cos η \cot θ,

where f_p = a_p, f_s = a_s exp(iη) and

w_{λ} = a_{λ}^{2} R_{λ}^{2} / (a_{p}^{2} R_{p}^{2} + a_{s}^{2} R_{s}^{2})

(where λ ∈ {p,s}) is the fractional energy contained in each polarization state.

As suggested by Figs. 2(a) and (c), the changes in R_λ introduced by the graphene layer are negligible. We therefore expect to observe no changes in the angular shifts Θ_GH and Θ_IF, as they are functions of R_λ solely. The spatial shifts Δ_GH and Δ_IF, on the other hand, contain a dependence on the phases ϕ_λ, and they are therefore affected by the presence of the graphene coating. Let us first discuss the IF shift. In this case ϕ_p − ϕ_s is very close to π [Figs. 2(b) and (d)], and the resulting spatial shift Δ_IF will be nonzero (but very small) even for linear polarization, in contrast to the case without graphene. The adimensional spatial IF shift is depicted in Fig. 3(a) and (b) for circular and linear polarized light, respectively.

Fig. 3 Spatial IF shift Δ_IF for (a) circular polarization and (b) 45° linear polarization for a graphene coated surface. The spatial IF shifts are nonzero even for linear polarization. Spatial GH shift Δ_GH for (c) p- polarization and (d) s-polarization for a graphene-coated surface. Since ∂ϕ_λ/∂θ ≠ 0, in both cases Δ_GH ≠ 0. In particular, since ϕ_p varies very rapidly with θ in the vicinity of the Brewster angle, the corresponding spatial shift for p-polarization [Panel (c)] is giant in modulus, and negative due to the fact that ϕ_p varies from 0 to −π [See Fig. 2(b)].

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More interesting is the case of the spatial GH shift. For an air-dielectric interface, one has ∂ϕ_λ/∂θ = 0 and therefore, according to Eq. (8a), Δ_GH = 0. It is in fact well known since the pioneering work of Goos and Hänchen [1], that Δ_GH = 0 occurs only in total internal reflection, where R_λ = 1 and ∂ϕ_λ/∂θ = 0. For the case of a graphene-coated surface, on the other hand, the phase ϕ_λ varies with θ for both s- and p-polarizations, as Figs. 2(b) and (d), respectively, show. In this case, then, we observe a nonzero spatial GH shift even without total internal reflection.

The adimensional spatial GH shift k₀Δ_GH occurring at a graphene-coated dielectric surface is depicted in Fig. 3(c) and (d) for p- and s-polarization, respectively. As can be seen, for both polarizations we have Δ_GH ≠ 0 although no total internal reflection takes place. In particular, ϕ_p varies very rapidly from 0 to −π in the vicinity of the Brewster angle θ_B. This corresponds to a giant and negative spatial GH shift. On the other hand, ϕ_s varies very smoothly with θ, thus resulting in a nonzero (but very small) spatial GH shift for s-polarization.

It is also instructive to calculate the GH shift for the case of total internal reflection (TIR). In this case, the reflection coefficients given by Eqs. (5) need to be modified to account for the fact that now the reflection takes place from the dielectric medium to air (where TIR can actually occur). Following the results of Ref. [44], the reflection coefficients for s–and p−polarization are then given by

r_{s} (θ) = \frac{n \cos θ - \sqrt{1 - n^{2} \sin^{2} θ} - n σ (Ω)}{n \cos θ + \sqrt{1 - n^{2} \sin^{2} θ} + n σ (Ω)},

r_{p} (θ) = \frac{\cos θ - n \sqrt{1 - n^{2} \sin^{2} θ} [1 - σ (Ω) \cos θ]}{n \cos θ + n \sqrt{1 - n^{2} \sin^{2} θ} [1 + σ (Ω) \cos θ]},

where θ is the incident angle, n is the refractive index of the dielectric medium and σ(Ω) is the graphene’s surface conductivity, as defined by Eq. (1). The reflection coefficients for internal reflection are depicted in Fig. 4.

Fig. 4 Modulus (left column) and phase (right column) of the reflection coefficients r_λ = R_λ exp(iϕ_λ) for p-polarization (top row) and s-polarization (bottom row) for the case with TIR. In all graphs, the solid black line corresponds to the case of the graphene-coated surface, while the green dashed curve corresponds to the case without graphene coating. The refractive index of the dielectric medium is chosen to be n = 1.5.

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The calculation of the GH and IF shifts for this case are essentially the same as explained above, with the only difference that upon TIR the reflection coefficients are close to 1 and vary slowly. According to Eqs. (8), this results in having θ_GH ≈ 0, θ_IF ≈ 0 for the angular and

k_{0} Δ_{GH} = w_{p} \frac{\partial ϕ_{p}}{\partial θ} + w_{s} \frac{\partial ϕ_{s}}{\partial θ},

k_{0} Δ_{IF} = - \frac{4 a_{p} a_{s}}{a_{p}^{2} + a_{s}^{2}} \cot θ \sin (η - \frac{ϕ_{p} - ϕ_{s}}{2}) \cos (\frac{ϕ_{p} - ϕ_{s}}{2}),

for the spatial shifts.

The adimensional spatial GH and IF shifts for TIR are depicted in Fig. 5. Again, the spatial shifts Δ_GH and Δ_IF contain a dependence on the phases ϕ_λ, and they are therefore affected by the presence of the graphene coating. The spatial IF is nonzero for both circular polarized light and for 45° linear polarized light. Above the critical angle the dependence is altered compared to the case without graphene and a negative shifts occurs even for linear polarized light. For the spatial GH shift, a giant negative shift is observed near the critical angle of the TIR for both, sand p-polarized light. Again, these shifts can be understood as a variation of the phases ϕ_λ.

Fig. 5 Spatial IF shift Δ_IF for (a) circular polarization and (b) 45° linear polarization for a graphene coated surface for the case with TIR. Spatial GH shift Δ_GH for (c) p- polarization and (d) s-polarization for a graphene-coated surface for the case with TIR.

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2. Conclusion

In conclusion, we have presented a detailed theoretical analysis of GH and IF shifts of a Gaussian beam impinging onto a graphene-coated dielectric surface. Our analysis revealed that the main effect of the graphene layer is to introduce, through its surface conductivity σ(ω), a dependence of the phases ϕ_λ of the reflection coefficients on the incident angle θ. This ultimately reflects in the appearance of a nonzero spatial GH and IF shifts. In particular a giant and negative spatial GH shift in the vicinity of the Brewster’s angle for p-polarization and additionally in the case with TIR at the critical angle for p- and s-polarization has been predicted. For the case of TIR, our calculations proved to be in agreement with the recently published experimental results [36].

Acknowledgments

The authors thank the German Ministry of Education and Science ( ZIK 03Z1HN31) for financial support.

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Goos-Hänchen and Imbert-Fedorov shifts for Gaussian beams impinging on graphene-coated surfaces

Abstract

1. Introduction

2. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (17)

Optics Express