Orbital angular momentum sidebands of vortex beams transmitted through a thin metamaterial slab

Wenguo Zhu; Heyuan Guan; Huihui Lu; Jieyuan Tang; Zhaihui Li; Jianhui Yu; Zhe Chen

doi:10.1364/OE.26.017378

1. Introduction

Reflection and transmission of plane waves at an interface between two media are described by the well-known Snell’s law and Fresnel equations, which, however, are not accurately complied by bounded beams [1,2]. Because of diffractive corrections, a bounded beam might be shifted in directions parallel and perpendicular to the plane of incidence, i.e., the so-called Goos–Hänchen (GH) and Imbert–Fedorov (IF) shifts, respectively [1,3]. The GH and IF shifts can be used for optical sensing [4,5] and precision metrology [6].

The GH and IF shifts of foundational Gaussian beam have been widely investigated [7–10], while these shifts for light beams carrying orbital angular momentum (OAM) have been less studied [11,12]. It was found that the OAM-dependent terms will appear in both the GH and IF shifts [13]. These OAM-dependent GH and IF shifts are labeled respectively as m-GH and m-IF shifts in this article. The m-GH and m-IF shifts result from the coupling between the complex vortex structure and the angular IF and angular GH shifts, respectively [13,14]. An OAM-dependent 4 × 4 matrix has been constructed to connect the GH and IF shifts of OAM-carrying beams with those of Gaussian beams [11]. The OAM-dependent shifts are accompanied by the deformation of OAM modes, which induces OAM sidebands. In 2012, the OAM sidebands of Laguerre-Gaussian (LG) beams were measured experimentally at an air-glass interface [15]. In general, the energies carried by sideband modes are small [15]. And the OAM-dependent shifts are tiny, which obstructs their applications in quantum information and precision metrology [13,16].

Epsilon-near-zero (ENZ) metamaterial has attracted significant attention owing to its applications ranging from the enhancement of directive emission [17,18], controlling of the electromagnetic flux [19], tailoring of light-matter interactions [20,21] to the enhancement nonlinear refractive index [22]. Recently, it has demonstrated that the ENZ metamaterial can enhance the GH shift [23] and spin splitting of Gaussian beams [24].

Here, we investigate the beam shifts of LG beams in photonic tunneling. Special attention is payed to the OAM-dependent term (m-GH) of the GH shift. The m-GH shift increases with the incident OAM, m, thus can enhance the GH shift of the transmitted beam. The giant m-GH shift can be obtained by the strong coupling among incident OAM {m} mode and “neighboring” sideband {m + 1} and {m-1} modes owing to the lack of the rotational symmetry of the involved optical system. The upper limit of the m-GH shift for LG beam is found to be |m|w₀/2(|m| + 1)^1/2 with w₀ being the incident beam waist. At the upper-limited m-GH shift, the beam shift of the transmitted beam is equal to this upper limit, since the other shifts vanish. We find further that by adjusting the incident linear polarization state, the m-GH shift of LG beams can approximate closely to its upper limit when transmitted through an ENZ metamaterial slab. The numerical simulation results based on finite-difference-time -domain (FDTD) method are presented and compared with the analytical ones.

2. Theory and model

It was demonstrated by Allen et al. that the high-order LG modes carry well-defined OAM [25]. The LG modes are labeled by two independent parameters: m and n, which correspond to the azimuthal and radial indices, respectively, and can be compactly referred to as $ϕ_{n}^{m}$ . To investigate the m-GH shift in photonic tunneling, we transmit a linearly polarized purely azimuthal LG mode, $E_{i} = [α_{p} | H 〉 + α_{s} | V 〉] | ϕ_{0}^{m} 〉$ , through a three-layer barrier structure, as shown in Fig. 1(a). The parameters α_p and α_s satisfy the relation |α_p|² + |α_s|² = 1, and $| H 〉$ and $| V 〉$ are the horizontal and vertical polarization states, being parallel and perpendicular to the plane of incidence, respectively. The global coordinate system is (x_g,y_g,z_g), while the local coordinate systems attached to the incident and transmitted beams are identical and denoted as (x, y, z). The LG beam incident obliquely on the barrier at an angle of θ. The incident beam can be considered as the superposition of plane waves that experience different modulations upon transmission [1]. Therefore, the distribution of transmitted light field will be changed. According to Refs [1,14], the angular spectrum of the transmitted beam is in form of

{\tilde{E}}_{t} = \sum_{γ = p, s} α_{γ} t_{γ} [| {\tilde{ϕ}}_{0}^{m} 〉 - (X_{γ} k_{x} + Y_{γ} k_{y}) / k_{d} | {\tilde{ϕ}}_{0}^{m} 〉] | Γ_{γ} 〉 .,

where k_d = 2πε_d^1/2/λ with λ and ε_d being the wavelength in vacuum and the dielectric permittivity, and the state

| Γ_{p, s} 〉

denotes

| H 〉

and

| V 〉

polarization states, respectively.

X_{p, s} = i t_{p, s}^{'} / t_{p, s},

Y_p_,_s = iα_s_,_pM/α_p_,_st_p_,_s, where M = (t_p-t_s)cotθ. X_p,s and Y_p,s are responsible for the conventional GH and IF shifts originated from Gaussian envelopes, respectively [5]. The IF shift can be associated with the Berry phases, k_ycotθ/k_d [1]. t_p_,_s are respectively the Fresnel transmission coefficients for the p and s waves, For an anisotropic metamaterial with a permittivity of ε_m = [ε_o, ε_o, ε_e], they are

t_{p} = \frac{4 k_{z}^{e} k_{z}^{d} ε_{o} ε_{d} \exp [i k_{z}^{e} d]}{{[k_{z}^{e} ε_{d} + k_{z}^{d} ε_{o}]}^{2} - {[k_{z}^{e} ε_{d} - k_{z}^{d} ε_{o}]}^{2} \exp [i 2 k_{z}^{e} d]},

t_{s} = \frac{4 k_{z}^{o} k_{z}^{d} \exp [i k_{z}^{o} d]}{{[k_{z}^{o} + k_{z}^{d}]}^{2} - {[k_{z}^{o} - k_{z}^{d}]}^{2} \exp [i 2 k_{z}^{o} d]},

where d is the thickness of metamaterial,

k_{z}^{d} = k_{d} \cos θ

,

k_{z}^{o} = k_{d} {[ε_{o} / ε_{d} - \cos^{2} θ]}^{1 / 2}

, and

k_{z}^{e} = k_{d} {[ε_{o} / ε_{d} - ε_{o} / ε_{e} \cos^{2} θ]}^{1 / 2}

, respectively.

t_{p, s}^{'}

are their derivatives with respect to incident angle θ. Making a Fourier transformation of its angular spectrum, the transmitted light field in spatial space can be obtained, which can be rewritten as a superposition of LG modes:

\begin{matrix} E_{t}^{} = \sum_{γ = p, s} α_{γ} t_{γ} {ϕ_{0}^{m} + \frac{θ_{0}}{2 \sqrt{2 (| m | + 1)}} [\sqrt{(| m | + 1)} Z_{γ}^{sgn [m]} | ϕ_{0}^{sgn [m] (| m | + 1)} 〉 - \\ \sqrt{| m |} Z_{γ}^{- sgn [m]} | ϕ_{0}^{sgn [m] (| m | - 1)} 〉 - Z_{γ}^{- sgn [m]} | ϕ_{1}^{sgn [m] (| m | - 1)} 〉]} | Γ_{γ} 〉, \end{matrix}

Here, θ₀ = 2(|m| + 1)^1/2/k_dw₀ with w₀ being the beam waist.

Z_{γ}^{sgn [m]} = X_{γ} - i sgn [m] Y_{γ}

with γ = p,s. One finds from Eq. (3) that the transmission induces the generation of new OAM modes. This can be understood as following: the inclination of the incident beam breaks the rotational symmetry (with respect to the z_g-axis) of the optical system containing light beams and metamaterial slab. During the photonic tunneling, the initial OAM mode (vortex structure) is perturbed; and new OAM modes (optical vortex) are born and coupled mutually. The transmitted beam contains many different OAM modes, as shown by the method section. Therefore, the transmitted light field can be rewritten as

E_{t}^{} = c_{p}^{m, 0, m', n'} | ϕ_{n'}^{m'} 〉 | H 〉 + c_{s}^{m, 0, m', n'} | ϕ_{n'}^{m'} 〉 | V 〉

, where

c_{p, s}^{m, 0, m', n'}

is the spatial Fresnel coefficients for p and s waves, describing the coupling among spatial modes [15]. The energies of each LG modes of the transmitted beam are calculated by the relation:

C_{m, 0}^{m', n'} = | c_{p, s}^{m, 0, m', n'} |^{2} + | c_{p, s}^{m, 0, m', n'} |^{2}

. When neglecting the second and higher-order terms with respect to θ₀, the transmitted light field is described by Eq. (3), indicating that the energies carried by initial {m} and neighboring sideband {sgn[m](|m|-1)} and {sgn[m](|m| + 1)} OAM modes are much larger than those by other OAM modes. That is transmission will induce a transformation of a LG

ϕ_{0}^{m}

mode mainly into a superposition of LG modes,

{ϕ_{0}^{m - 1}, ϕ_{0}^{m}, ϕ_{0}^{m + 1}, ϕ_{1}^{sgn [m] (| m | - 1)}}

, as shown in Fig. 1(b) [15]. The OAM {sgn[m](|m|-1)} mode contains two different LG modes with identical azimuthal index. According to Eq. (3),

C_{m, 0}^{sgn [m] (| m | - 1), 0} = m C_{m, 0}^{sgn [m] (| m | - 1), 1}

and

C_{m, 0}^{sgn [m] (| m | + 1), 0} = C_{m, 0}^{sgn [m] (| m | - 1), 1}

+ C_{m, 0}^{sgn [m] (| m | - 1), 0} .

The energies of “neighboring” sideband OAM modes are always identical.

Fig. 1 (a) The centroid of a vortex beam will shift along x axis when transmitted through a three-layer barrier structure. (b) When transmitted through the three-layer barrier structure, a linearly polarized LG beam $| ϕ_{0}^{3} 〉$ will be distorted, and the transmitted beam can be considered as a superposition of LG $| ϕ_{0}^{4} 〉$ , $| ϕ_{0}^{3} 〉$ , $| ϕ_{0}^{2} 〉$ , and $| ϕ_{1}^{2} 〉$ modes.

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As a superposition of LG modes, the transmitted beam might lose circular symmetry, and thus the beam centroid may shift [1,26]. The displacements of the centroids of the transmitted beam along the x axis, with respect to the predictions of geometrical optics, is defined as $Δ X = \iint x | E_{t}^{} |^{2} d x d y / \iint | E_{t}^{} |^{2} d x d y$ [1]. After some straight calculation, we have

Δ X = \sum_{γ = p, s} | α_{γ} t_{γ} |^{2} [Re (X_{γ}) + m Im (Y_{γ})] / (k_{0} W),

where the total energy of the transmitted beam is

W = \sum_{γ = p, s} | α_{γ} t_{γ} |^{2} {1 + θ_{0}^{2} [| X_{γ} |^{2} + | Y_{γ} |^{2}] / 4},

The first term of Eq. (4) corresponds to the conventional spatial GH shift originated from the Gaussian envelope, which are independent of the incident OAM [1]; while the second term is m-GH shift, which is OAM-dependent, existing only for higher-order LG modes. The m-GH shift (denoted as ∆X_m in the following) originates from the coupling between the complex vortex structure with the angular IF shift, Im(Y_η) [13]. The m-GH shifts ∆X_m vanishes if the incident beam is purely horizontally or vertically polarized.

To investigate OAM-dependent GH shift, we introduce parameters β = |α_s/α_p| and δ = arg[α_s/α_p], which are the amplitude ratio and phase difference between $| V 〉$ and $| H 〉$ polarization components of the incident beam, respectively. Thus Eq. (4) can be rewritten as

Δ X_{m} = \frac{1}{k_{d}} \frac{β \cos δ [| t_{p} |^{2} - | t_{s} |^{2}] \cot θ}{| t_{p} |^{2} + β^{2} | t_{s} |^{2} + \frac{θ_{0}^{2}}{4} [| t_{p}^{'} |^{2} + β^{2} | t_{s}^{'} |^{2} + (1 + β^{2}) | M |^{2}]} .

The ∆X_m changes with the phase difference δ according to cosine function. ∆X_m is maximized when δ = 0 and π. Therefore, the incident beam should be linearly polarized. Here, we set δ = 0.

For a given incident angle, the m-GH shift changes with initial amplitude ratio β, and two peak values can be found among all β. Therefore, the two peaks of m-GH shifts are dependent on the incident angle and are governed by

Δ X_{m, p k}^{\pm} = \frac{\pm m (| t_{p} |^{2} - | t_{s} |^{2}) \cot θ}{2 k_{d} \sqrt{| t_{p} |^{2} + θ_{0}^{2} [| t_{p}^{'} |^{2} + | M |^{2}] / 4} \sqrt{| t_{s} |^{2} + θ_{0}^{2} [| t_{s}^{'} |^{2} + | M |^{2}] / 4}},

The two peaks of m-GH shifts are obtained respectively in the positive and negative β regions:

β_{p k}^{\pm} (θ) = \pm \sqrt{\frac{| t_{p} |^{2} + θ_{0}^{2} [| t_{p}^{'} |^{2} + | M |^{2}] / 4}{| t_{s} |^{2} + θ_{0}^{2} [| t_{s}^{'} |^{2} + | M |^{2}] / 4}} .

The absolute value of m-GH shift is smaller than Δ_up = |m|w₀ /2(|m| + 1)^1/2. For the case of |t_p| = 0, |t_s|≠0, Eq. (7) becomes

Δ X_{m, p k}^{\pm} \approx \frac{\mp m w_{0}}{2 \sqrt{| m | + 1}} \frac{1}{\sqrt{1 + | t_{p}^{'} / t_{s} \cot θ |^{2}}} .

When

| t_{p}^{'} |^{2} < < | t_{s} \cot θ |^{2}

,

Δ X_{m, p k}^{\pm} \to \mp m w_{0} / 2 {(| m | + 1)}^{1 / 2}

. For the case of |t_s| = 0, same results can be obtained by making the replacements of t_p_,_s →t_s_,_p and

t_{p, s}^{'} \to t_{s, p}^{'}

. Therefore, Δ_up = |m|w₀ /2(|m| + 1)^1/2 is the upper limit of the m-GH shift, which determined only by the incident beam waist and incident OAM. It is worth noted that at the upper-limited m-GH shift, the spatial GH shift of the transmitted beam is equal to the m-GH shift owing to the vanishment of the first term in Eq. (4).

The giant beam shifts of the transmitted beams result from the strong coupling among OAM modes in barrier medium. For the upper-limited m-GH shift with |t_p| = 0 and δ = 0, and $β = β_{p k}^{+}$ , the transmitted beam can be written in the following form

\begin{matrix} E_{t}^{} = \frac{α_{p}}{k_{d}^{} w_{0}^{}} {| t_{s} \cot θ | \sqrt{| m | + 1} | ϕ_{0}^{m} 〉 - sgn [m] t_{s} \cot θ [\sqrt{| m | + 1} \\ | ϕ_{0}^{sgn [m] (| m + 1)} 〉 + \sqrt{| m |} | ϕ_{0}^{sgn [m] (| m | - 1)} 〉 + | ϕ_{1}^{sgn [m] (| m | - 1)} 〉] / \sqrt{2}} | V 〉 . \end{matrix}

Strangely, the transmitted beam is homogenously polarized. This is because that the energy carried by the

| H 〉

polarization component is much smaller than that by

| V 〉

polarization component, since |t_p| = 0 and

| t_{p}^{'} |<<| t_{s} \cot θ |^{2}

. From Eq. (10) one finds that, the intensity profile of the transmitted beam is strongly deformed comparing to the incident intensity profile, resulting in large OAM-dependent shifts. The strong deformations of intensity profiles are also found in the reflected Gaussian beams after postselected by an analyzer [27,28]. The energies of the LG modes have following relationship:

C_{m, 0}^{m, 0} = 2 C_{m, 0}^{sgn [m] (| m | + 1), 0} = 2 [C_{m, 0}^{sgn [m] (| m | - 1), 1} + C_{m, 0}^{sgn [m] (| m | - 1), 0}] .

The energies of two “neighboring” sideband {m + 1} and {m-1} OAM mode are identical, which are equal to half energy of the {m} mode. Therefore, three OAM modes interfere efficiently, which leads to the asymmetric change of the intensity profile and thus the upper-limited OAM-dependent shift.

These results are universal since no media are specified in the above discussion. They can be extended to the cases of reflection. In the following, we will design a ENZ metamaterial for the realization of the upper limit for the m-GH shift.

3. Result and discussion

The m-GH shift may exist in air-dielectric interface and other configurations [4,7,23]. However, in order to obtain the upper-limited m-GH shift, the Fresnel transmission coefficients should satisfy several conditions, as shown above. The regular materials such as dielectric and metal can hardly achieve these conditions simultaneously [14]. However, the Fresnel transmission coefficients of a thin ENZ metamaterial slab can be flexibility controlled by tuning the slab thickness and the incident angle [29], therefore, the upper-limited m-GH shift could be possibly obtained.

An anisotropy ENZ metamaterial is chosen as the barrier medium in the three-layer structure, whose permittivity is ε_m = [ε_o, ε_o, ε_e] with ε_e being near zero. This kind of metamaterial can be formed by embedding arrays of parallel metallic wires in a host dielectric material [30]. ε_o is determined by the permittivity of host material, which is equal to 2.4 for Al₂O₃ near 633nm. ε_e can take positive and negative values by adjusting the wire radius and period [31]. Here, we set ε_e = 0.01 + 0.01i, and set the permittivity of the upper and lower layers to be ε_d = 1 for simplicity.

Figure 2(a) shows the normalized m-GH shifts as functions of the incident angle θ and initial amplitude ratio β for m = 1.The m-GH shift ∆X_m can be well controlled by β. ∆X_m changes sign when β cross zero point, where ∆X_m = 0. In the range of θ>10°, the m-GH shift can always trend to its upper limit by optimizing parameter β. When θ = 10° and 30°, the peak values are $Δ X_{m, p k}^{\pm}$ = $\mp$ 0.982Δ_up and $\mp$ 0.995Δ_up. For a larger incident angle, a smaller |β| is required for the achievement of the upper limit, as shown by Fig. 2(b), where the normalized GH shifts of the transmitted beam, ΔX/Δ_up [Eq. (4)], are plotted for m = ± 1 when θ = 5°, 10°, 30°. When θ = 10° and 30°, the peak positions are $β_{p k}^{\pm}$ = ± 0.052 and ± 0.015, respectively.

Fig. 2 (a) The normalized m-GH shift ΔX_m/Δ_up changing with the incident angle θ and initial amplitude ratio β when m = 1. (b) The m-GH shift ΔX_m/Δ_up changing with β for m = ± 1 when θ = 5° (red), 10° (blue), 30° (green), respectively.

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Upon transmission, the OAM {m} mode will couple with the sideband {m-1} and {m + 1} modes. With the increase of the thickness of the ENZ metamaterial d, the normalized energy of the {m} mode of the transmitted beam decreases gradually, while the sideband {m-1} and {m + 1} modes increase, as shown by Fig. 3(b), where the normalized energies of different LG modes are plotted for θ = 10°, β = −0.07, m = 3. The normalized energies of {m}, {m-1}, and {m + 1} modes tend to asymptotic values of 50%, 25%, and 25%, respectively. Therefore, the relationship (11) is almost satisfied. The m-GH shift of the transmitted beam should approximate its upper limit value, which is clear seen in Fig. 3(a). It should be noted that, although the GH shift of the transmitted beam is investigated, the OAM sidebands associate with both the GH and IF shifts since any deformations of light field will induce the OAM sidebands.

Fig. 3 The normalized m-GH shifts ΔX_m/Δ_up (a) and energies of LG modes of the transmitted beam (b) changing with the thickness of ENZ metamaterial d when θ = 10°, β = −0.07, m = 3. The energies of different LG modes normalized by the total energy of the transmitted beam.

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The spatial GH shift of the transmitted LG beams, ∆X, vary with incident OAM m owing to their OAM-dependent terms. Figure 4(a) shows the dependences of ∆X on m for the cases of θ = 6°, β = 0 (red circles), θ = 10°, β = −0.052 (blue circles), and θ = 6°, β = 0.3 (green circles), respectively. When m = 0, although the m-GH shift vanish, ∆X is nonzero because the first term in Eq. (4) is nonzero. However, this term is smaller than a wavelength λ. In the cases of θ = 6°, β = 0.3, ∆X is linearly proportional to the incident OAM m. However, when θ = 10°, β = −0.052, ∆X changes nonlinearly with m. The parameters θ = 10° and β = −0.052 are optimized parameters for the m-GH shift at m = ± 1. The upper limit of m-GH shift at m = ± 1 is 8.7λ. For 1 ≤|m|≤5, the m-GH shift ∆X_m are always larger than 0.86Δ_up. The giant m-GH shift are clearly shown in Fig. 4(b), where the intensity distributions of the transmitted beam along x axis for different m are plotted.

Fig. 4 (a) The spatial GH shift ∆X shift changing with incident OAM m when θ = 6°, β = 0 (red circles), θ = 6°, β = 0.3 (green circles), and θ = 10°, β = −0.052 (blue circles), respectively. (b) The intensity profiles of the transmitted beam along x axis for m = −5:5, when θ = 10°, β = −0.052.

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Upon transmission, the incident OAM mode will convert partly to the “neighboring” OAM modes. The OAM spectra of the transmitted beams are shown in Fig. 5 when θ = 10° and β = −0.05. The energies of the “neighboring” OAM modes, $C_{m, 0}^{sgn [m] (| m | + 1), 0}$ and $C_{m, 0}^{sgn [m] (| m | - 1), 1} + C_{m, 0}^{sgn [m] (| m | - 1), 0}$ are identical for all incident OAM m. The energies of {m} mode $C_{m, 0}^{m, 0}$ decrease with the increase of |m|. The relationship (11) can only be satisfied when m = ± 1. Therefore, the OAM-dependent shifts can reach their upper limits only for the cases of m = ± 1.

Fig. 5 The energies of OAM modes of the transmitted beam for different incident OAM m when θ = 10° and β = −0.052. (a) Three-dimensional bar graph, (b) Two-dimensional pseudocolor plot.

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We performed full-wave numerical simulations to our three-layer barrier structure with ENZ metamaterial to show the OAM-dependent shifts intuitively, as a verification of our theoretical predictions above. The simulations were finished via commercial FDTD software [32]. In the simulations, the incident beam waist is set to be 7.9λ, and the simulation region spans 40λ × 40λ × 12λ along x_g, y_g, and z_g axes.

When the incident angle is θ = 8°, the giant m-GH shift occurs, thus the transmitted beams undergo OAM-dependent shifts along x_g axis, as shown in Fig. 6(a) and(b). The GH shifts of the transmitted beams, ∆X, change with the initial amplitude ratio β. When β = ± 0.2, ∆X is at their peaks, and the m-GH shift |∆X_m| reach its upper limit of 2.79λ. The simulation results are consistent with the analytical predictions (by Eq. (3)).

Fig. 6 The Numerical verification of the theoretically predicted m-GH shifts. The normalized intensity distributions in x_gz_g plane for m = 1, and β = 0.2 (a) and −0.2 (b), respectively. (c) The comparison of the analytical results (solid lines) and numerical results (circles) on the GH shifts ΔX. The two insets in (c) are the intensity distributions in xy plane for θ = 8° and β = ± 0.2, respectively.

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4. Conclusions

We have demonstrated theoretically and numerically the giant m-GH shifts of LG beams transmitted through a three-layer barrier structure. The upper limits of the m-GH shifts are predicted theoretically, which rely simply on the incident OAM and beam waist, |m|w₀/2(|m| + 1)^1/2. To obtain the upper-limited shifts, the incident OAM {m} mode should couple strongly with the “neighboring” sideband {m + 1} and {m-1} modes, and their energies should satisfy relationship (11). An anisotropy ENZ metamaterial is used to enhance the OAM-dependent beam shifts. By modulating the linear polarization state, the m-GH shift can approximate closely to its upper limit. These findings provide a deeper understanding of the interaction between metamaterials and OAM-carried light field and thereby facilitate the OAM-based applications.

Funding

National Natural Science Foundation of China (61705086, 61505069, 61675092, 61475066, 61575084); Natural Science Foundation of Guangdong Province (2017A030313375, 2017A010102006, 2016A030311019, 2016TQ03X962), Science and technology projects of Guangdong Province (2016A030313079, 2016A030311019)

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Orbital angular momentum sidebands of vortex beams transmitted through a thin metamaterial slab

Abstract

1. Introduction

2. Theory and model

3. Result and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (6)

Equations (12)

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