Abstract

A pure vortex beam carrying m-order orbital angular momentum (OAM) will be deformed when transmitting through a thin slab, and “neighboring” sideband {m + 1} and {m-1} modes will emerge. The emergence of the OAM sideband is accompanied with OAM-dependent Goos–Hänchen (GH) shift. When the energies carried by the {m} mode of the transmitted beam and by the sideband modes are identical, the OAM-dependent shifts reach their upper limits, |m|w0/2(|m| + 1)1/2, where w0 is the incident beam waist. The epsilon-near-zero metamaterial is found to be suitable to achieve the upper-limited OAM-dependent GH shifts. These findings provide a deeper insight into the beam shifts of vortex beams and have potential applications in the optical sensing, detection of OAM, and other OAM-based applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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 ϕnm. To investigate the m-GH shift in photonic tunneling, we transmit a linearly polarized purely azimuthal LG mode, Ei=[αp|H+αs|V]|ϕ0m, through a three-layer barrier structure, as shown in Fig. 1(a). The parameters αp and αs satisfy the relation |αp|2 + |αs|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 (xg,yg,zg), 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

E˜t=γ=p,sαγtγ[|ϕ˜0m(Xγkx+Yγky)/kd|ϕ˜0m]|Γγ.,
where kd = 2πεd1/2/λ with λ and εd being the wavelength in vacuum and the dielectric permittivity, and the state |Γp,s denotes |H and |Vpolarization states, respectively. Xp,s=itp,s'/tp,s, Yp,s = s,pM/αp,stp,s, where M = (tp-ts)cotθ. Xp,s and Yp,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, kycotθ/kd [1]. tp,s are respectively the Fresnel transmission coefficients for the p and s waves, For an anisotropic metamaterial with a permittivity of εm = [εo, εo, εe], they are
tp=4kzekzdεoεdexp[ikzed][kzeεd+kzdεo]2[kzeεdkzdεo]2exp[i2kzed],
ts=4kzokzdexp[ikzod][kzo+kzd]2[kzokzd]2exp[i2kzod],
where d is the thickness of metamaterial,kzd=kdcosθ, kzo=kd[εo/εdcos2θ]1/2, and kze=kd[εo/εdεo/εecos2θ]1/2, respectively. tp,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:
Et=γ=p,sαγtγ{ϕ0m+θ022(|m|+1)[(|m|+1)Zγsgn[m]|ϕ0sgn[m](|m|+1)|m|Zγsgn[m]|ϕ0sgn[m](|m|1)Zγsgn[m]|ϕ1sgn[m](|m|1)]}|Γγ,
Here, θ0 = 2(|m| + 1)1/2/kdw0 with w0 being the beam waist. Zγsgn[m]=Xγisgn[m]Yγ 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 zg-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 Et=cpm,0,m',n'|ϕn'm'|H+csm,0,m',n'|ϕn'm'|V, where cp,sm,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: Cm,0m',n'=|cp,sm,0,m',n'|2+|cp,sm,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 ϕ0m mode mainly into a superposition of LG modes, {ϕ0m1,ϕ0m,ϕ0m+1,ϕ1sgn[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), Cm,0sgn[m](|m|1),0=mCm,0sgn[m](|m|1),1 and Cm,0sgn[m](|m|+1),0=Cm,0sgn[m](|m|1),1 +Cm,0sgn[m](|m|1),0.The energies of “neighboring” sideband OAM modes are always identical.

 figure: Fig. 1

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 |ϕ03will be distorted, and the transmitted beam can be considered as a superposition of LG |ϕ04, |ϕ03, |ϕ02, and |ϕ12 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 ΔX=x|Et|2dxdy/|Et|2dxdy [1]. After some straight calculation, we have

ΔX=γ=p,s|αγtγ|2[Re(Xγ)+mIm(Yγ)]/(k0W),
where the total energy of the transmitted beam is
W=γ=p,s|αγtγ|2{1+θ02[|Xγ|2+|Yγ|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 ∆Xm in the following) originates from the coupling between the complex vortex structure with the angular IF shift, Im(Yη) [13]. The m-GH shifts ∆Xm vanishes if the incident beam is purely horizontally or vertically polarized.

To investigate OAM-dependent GH shift, we introduce parameters β = |αs/αp| and δ = arg[αs/αp], 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

ΔXm=1kdβcosδ[|tp|2|ts|2]cotθ|tp|2+β2|ts|2+θ024[|tp'|2+β2|ts'|2+(1+β2)|M|2].
The ∆Xm changes with the phase difference δ according to cosine function. ∆Xm 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

ΔXm,pk±=±m(|tp|2|ts|2)cotθ2kd|tp|2+θ02[|tp'|2+|M|2]/4|ts|2+θ02[|ts'|2+|M|2]/4,
The two peaks of m-GH shifts are obtained respectively in the positive and negative β regions:
βpk±(θ)=±|tp|2+θ02[|tp'|2+|M|2]/4|ts|2+θ02[|ts'|2+|M|2]/4.
The absolute value of m-GH shift is smaller than Δup = |m|w0 /2(|m| + 1)1/2. For the case of |tp| = 0, |ts|≠0, Eq. (7) becomes
ΔXm,pk±mw02|m|+111+|tp'/tscotθ|2.
When|tp'|2<<|tscotθ|2, ΔXm,pk±mw0/2(|m|+1)1/2. For the case of |ts| = 0, same results can be obtained by making the replacements of tp,sts,p and tp,s'ts,p'. Therefore, Δup = |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 |tp| = 0 and δ = 0, and β=βpk+, the transmitted beam can be written in the following form

Et=αpkdw0{|tscotθ||m|+1|ϕ0msgn[m]tscotθ[|m|+1|ϕ0sgn[m](|m+1)+|m||ϕ0sgn[m](|m|1)+|ϕ1sgn[m](|m|1)]/2}|V.
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 |tp| = 0 and |tp'|<<|tscotθ|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:
Cm,0m,0=2Cm,0sgn[m](|m|+1),0=2[Cm,0sgn[m](|m|1),1+Cm,0sgn[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 εm = [εo, εo, εe] with εe being near zero. This kind of metamaterial can be formed by embedding arrays of parallel metallic wires in a host dielectric material [30]. εo is determined by the permittivity of host material, which is equal to 2.4 for Al2O3 near 633nm. εe can take positive and negative values by adjusting the wire radius and period [31]. Here, we set εe = 0.01 + 0.01i, and set the permittivity of the upper and lower layers to be εd = 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 ∆Xm can be well controlled by β. ∆Xm changes sign when β cross zero point, where ∆Xm = 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 ΔXm,pk± = 0.982Δup and 0.995Δup. 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, ΔXup [Eq. (4)], are plotted for m = ± 1 when θ = 5°, 10°, 30°. When θ = 10° and 30°, the peak positions are βpk± = ± 0.052 and ± 0.015, respectively.

 figure: Fig. 2

Fig. 2 (a) The normalized m-GH shift ΔXmup changing with the incident angle θ and initial amplitude ratio β when m = 1. (b) The m-GH shift ΔXmup 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.

 figure: Fig. 3

Fig. 3 The normalized m-GH shifts ΔXmup (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 ∆Xm are always larger than 0.86Δup. 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.

 figure: Fig. 4

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, Cm,0sgn[m](|m|+1),0 and Cm,0sgn[m](|m|1),1+Cm,0sgn[m](|m|1),0 are identical for all incident OAM m. The energies of {m} mode Cm,0m,0decrease 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.

 figure: Fig. 5

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 xg, yg, and zg axes.

When the incident angle is θ = 8°, the giant m-GH shift occurs, thus the transmitted beams undergo OAM-dependent shifts along xg 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 |∆Xm| reach its upper limit of 2.79λ. The simulation results are consistent with the analytical predictions (by Eq. (3)).

 figure: Fig. 6

Fig. 6 The Numerical verification of the theoretically predicted m-GH shifts. The normalized intensity distributions in xgzg 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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References

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  1. K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
    [Crossref]
  2. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
    [Crossref] [PubMed]
  3. J. B. Götte and M. R. Dennis, “Generalized shifts and weak values for polarization components of reflected light beams,” New J. Phys. 14(7), 073016 (2012).
    [Crossref]
  4. X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
    [Crossref] [PubMed]
  5. X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41(17), 4032–4035 (2016).
    [Crossref] [PubMed]
  6. X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
    [Crossref] [PubMed]
  7. S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
    [Crossref]
  8. W. Zhu, M. Jiang, H. Guan, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Tunable spin splitting of Laguerre–Gaussian beams in graphene metamaterials,” Photon. Res. 5(6), 684 (2017).
    [Crossref]
  9. Y. Xiang, X. Jiang, Q. You, J. Guo, and X. Dai, “Enhanced spin Hall effect of reflected light with guided-wave surface plasmon resonance,” Photon. Res. 5(5), 467–472 (2017).
    [Crossref]
  10. T. Tang, C. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6(1), 30762 (2016).
    [Crossref] [PubMed]
  11. W. Zhu, L. Zhuo, M. Jiang, H. Guan, J. Yu, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Controllable symmetric and asymmetric spin splitting of Laguerre-Gaussian beams assisted by surface plasmon resonance,” Opt. Lett. 42(23), 4869–4872 (2017).
    [Crossref] [PubMed]
  12. C. Prajapati, “Numerical calculation of beam shifts for higher-order Laguerre-Gaussian beams upon transmission,” Opt. Commun. 389, 290–296 (2017).
    [Crossref]
  13. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
    [Crossref]
  14. Z. Xiao, H. Luo, and S. Wen, “Goos-Hanchen and Imbert-Fedorov shifts of vortex beams at air left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
    [Crossref]
  15. W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
    [Crossref] [PubMed]
  16. J. Qiu, C. Ren, and Z. Zhang, “Precisely measuring the orbital angular momentum of beams via weak measurement,” Phys. Rev. A 93(6), 063841 (2016).
    [Crossref]
  17. I. Liberal and N. Engheta, “Near-zero refractive index photonics,” Nat. Photonics 11(3), 149–158 (2017).
    [Crossref]
  18. X.-T. He, Y.-N. Zhong, Y. Zhou, Z. C. Zhong, and J. W. Dong, “Dirac directional emission in anisotropic zero refractive index photonic crystals,” Sci. Rep. 5(1), 13085 (2015).
    [Crossref] [PubMed]
  19. Y. Li, H. T. Jiang, W. W. Liu, J. Ran, Y. Lai, and H. Chen, “Experimental realization of subwavelength flux manipulation in anisotropic near-zero index metamaterials,” Europhys. Lett. 113(5), 57006 (2016).
    [Crossref]
  20. I. Liberal, A. M. Mahmoud, Y. Li, B. Edwards, and N. Engheta, “Photonic doping of epsilon-near-zero media,” Science 355(6329), 1058–1062 (2017).
    [Crossref] [PubMed]
  21. S. Campione, S. Liu, A. Benz, J. F. Klem, M. B. Sinclair, and I. Brener, “Epsilon-near-zero modes for tailored light-matter interaction,” Phys. Rev. Appl. 4(4), 044011 (2015).
    [Crossref]
  22. M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
    [Crossref] [PubMed]
  23. Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
    [Crossref] [PubMed]
  24. W. Zhu and W. She, “Enhanced spin Hall effect of transmitted light through a thin epsilon-near-zero slab,” Opt. Lett. 40(13), 2961–2964 (2015).
    [Crossref] [PubMed]
  25. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref] [PubMed]
  26. J. Zhang, X. X. Zhou, X. H. Ling, S. Z. Chen, H. L. Luo, and S. C. Wen, “Orbit-orbit interaction and photonic orbital Hall effect in reflection of a light beam,” Chin. Phys. B 23(6), 064215 (2014).
    [Crossref]
  27. J. B. Götte and M. R. Dennis, “Limits to superweak amplification of beam shifts,” Opt. Lett. 38(13), 2295–2297 (2013).
    [Crossref] [PubMed]
  28. A. Aiello, M. Merano, and J. P. Woerdman, “Brewster cross polarization,” Opt. Lett. 34(8), 1207–1209 (2009).
    [Crossref] [PubMed]
  29. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
    [Crossref]
  30. J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
    [Crossref] [PubMed]
  31. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2014).
    [Crossref]
  32. Lumerical FDTD Solutions, www.lumerical.com

2018 (1)

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref] [PubMed]

2017 (8)

X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
[Crossref] [PubMed]

W. Zhu, M. Jiang, H. Guan, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Tunable spin splitting of Laguerre–Gaussian beams in graphene metamaterials,” Photon. Res. 5(6), 684 (2017).
[Crossref]

I. Liberal and N. Engheta, “Near-zero refractive index photonics,” Nat. Photonics 11(3), 149–158 (2017).
[Crossref]

C. Prajapati, “Numerical calculation of beam shifts for higher-order Laguerre-Gaussian beams upon transmission,” Opt. Commun. 389, 290–296 (2017).
[Crossref]

W. Zhu, L. Zhuo, M. Jiang, H. Guan, J. Yu, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Controllable symmetric and asymmetric spin splitting of Laguerre-Gaussian beams assisted by surface plasmon resonance,” Opt. Lett. 42(23), 4869–4872 (2017).
[Crossref] [PubMed]

I. Liberal, A. M. Mahmoud, Y. Li, B. Edwards, and N. Engheta, “Photonic doping of epsilon-near-zero media,” Science 355(6329), 1058–1062 (2017).
[Crossref] [PubMed]

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Y. Xiang, X. Jiang, Q. You, J. Guo, and X. Dai, “Enhanced spin Hall effect of reflected light with guided-wave surface plasmon resonance,” Photon. Res. 5(5), 467–472 (2017).
[Crossref]

2016 (5)

M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
[Crossref] [PubMed]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41(17), 4032–4035 (2016).
[Crossref] [PubMed]

Y. Li, H. T. Jiang, W. W. Liu, J. Ran, Y. Lai, and H. Chen, “Experimental realization of subwavelength flux manipulation in anisotropic near-zero index metamaterials,” Europhys. Lett. 113(5), 57006 (2016).
[Crossref]

J. Qiu, C. Ren, and Z. Zhang, “Precisely measuring the orbital angular momentum of beams via weak measurement,” Phys. Rev. A 93(6), 063841 (2016).
[Crossref]

T. Tang, C. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6(1), 30762 (2016).
[Crossref] [PubMed]

2015 (4)

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
[Crossref] [PubMed]

X.-T. He, Y.-N. Zhong, Y. Zhou, Z. C. Zhong, and J. W. Dong, “Dirac directional emission in anisotropic zero refractive index photonic crystals,” Sci. Rep. 5(1), 13085 (2015).
[Crossref] [PubMed]

S. Campione, S. Liu, A. Benz, J. F. Klem, M. B. Sinclair, and I. Brener, “Epsilon-near-zero modes for tailored light-matter interaction,” Phys. Rev. Appl. 4(4), 044011 (2015).
[Crossref]

W. Zhu and W. She, “Enhanced spin Hall effect of transmitted light through a thin epsilon-near-zero slab,” Opt. Lett. 40(13), 2961–2964 (2015).
[Crossref] [PubMed]

2014 (2)

J. Zhang, X. X. Zhou, X. H. Ling, S. Z. Chen, H. L. Luo, and S. C. Wen, “Orbit-orbit interaction and photonic orbital Hall effect in reflection of a light beam,” Chin. Phys. B 23(6), 064215 (2014).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2014).
[Crossref]

2013 (2)

J. B. Götte and M. R. Dennis, “Limits to superweak amplification of beam shifts,” Opt. Lett. 38(13), 2295–2297 (2013).
[Crossref] [PubMed]

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

2012 (3)

W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
[Crossref] [PubMed]

J. B. Götte and M. R. Dennis, “Generalized shifts and weak values for polarization components of reflected light beams,” New J. Phys. 14(7), 073016 (2012).
[Crossref]

Z. Xiao, H. Luo, and S. Wen, “Goos-Hanchen and Imbert-Fedorov shifts of vortex beams at air left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
[Crossref]

2010 (1)

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

2009 (1)

2008 (2)

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
[Crossref] [PubMed]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

2007 (1)

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Aiello, A.

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
[Crossref] [PubMed]

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

A. Aiello, M. Merano, and J. P. Woerdman, “Brewster cross polarization,” Opt. Lett. 34(8), 1207–1209 (2009).
[Crossref] [PubMed]

Alam, M. Z.

M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
[Crossref] [PubMed]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Alù, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Bartal, G.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2014).
[Crossref]

Benz, A.

S. Campione, S. Liu, A. Benz, J. F. Klem, M. B. Sinclair, and I. Brener, “Epsilon-near-zero modes for tailored light-matter interaction,” Phys. Rev. Appl. 4(4), 044011 (2015).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

Boyd, R. W.

M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
[Crossref] [PubMed]

Brener, I.

S. Campione, S. Liu, A. Benz, J. F. Klem, M. B. Sinclair, and I. Brener, “Epsilon-near-zero modes for tailored light-matter interaction,” Phys. Rev. Appl. 4(4), 044011 (2015).
[Crossref]

Cai, L.

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Campione, S.

S. Campione, S. Liu, A. Benz, J. F. Klem, M. B. Sinclair, and I. Brener, “Epsilon-near-zero modes for tailored light-matter interaction,” Phys. Rev. Appl. 4(4), 044011 (2015).
[Crossref]

Chan, C. T.

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
[Crossref] [PubMed]

Chen, H.

Y. Li, H. T. Jiang, W. W. Liu, J. Ran, Y. Lai, and H. Chen, “Experimental realization of subwavelength flux manipulation in anisotropic near-zero index metamaterials,” Europhys. Lett. 113(5), 57006 (2016).
[Crossref]

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
[Crossref] [PubMed]

Chen, S.

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Chen, S. Z.

J. Zhang, X. X. Zhou, X. H. Ling, S. Z. Chen, H. L. Luo, and S. C. Wen, “Orbit-orbit interaction and photonic orbital Hall effect in reflection of a light beam,” Chin. Phys. B 23(6), 064215 (2014).
[Crossref]

Chen, Z.

Dai, X.

De Leon, I.

M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
[Crossref] [PubMed]

Dennis, M. R.

J. B. Götte and M. R. Dennis, “Limits to superweak amplification of beam shifts,” Opt. Lett. 38(13), 2295–2297 (2013).
[Crossref] [PubMed]

J. B. Götte and M. R. Dennis, “Generalized shifts and weak values for polarization components of reflected light beams,” New J. Phys. 14(7), 073016 (2012).
[Crossref]

Dong, J. W.

X.-T. He, Y.-N. Zhong, Y. Zhou, Z. C. Zhong, and J. W. Dong, “Dirac directional emission in anisotropic zero refractive index photonic crystals,” Sci. Rep. 5(1), 13085 (2015).
[Crossref] [PubMed]

Du, J.

Edwards, B.

I. Liberal, A. M. Mahmoud, Y. Li, B. Edwards, and N. Engheta, “Photonic doping of epsilon-near-zero media,” Science 355(6329), 1058–1062 (2017).
[Crossref] [PubMed]

Engheta, N.

I. Liberal, A. M. Mahmoud, Y. Li, B. Edwards, and N. Engheta, “Photonic doping of epsilon-near-zero media,” Science 355(6329), 1058–1062 (2017).
[Crossref] [PubMed]

I. Liberal and N. Engheta, “Near-zero refractive index photonics,” Nat. Photonics 11(3), 149–158 (2017).
[Crossref]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Götte, J. B.

J. B. Götte and M. R. Dennis, “Limits to superweak amplification of beam shifts,” Opt. Lett. 38(13), 2295–2297 (2013).
[Crossref] [PubMed]

J. B. Götte and M. R. Dennis, “Generalized shifts and weak values for polarization components of reflected light beams,” New J. Phys. 14(7), 073016 (2012).
[Crossref]

Guan, H.

Guo, J.

He, X.-T.

X.-T. He, Y.-N. Zhong, Y. Zhou, Z. C. Zhong, and J. W. Dong, “Dirac directional emission in anisotropic zero refractive index photonic crystals,” Sci. Rep. 5(1), 13085 (2015).
[Crossref] [PubMed]

Hermosa, N.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
[Crossref] [PubMed]

Huang, K.

X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
[Crossref] [PubMed]

Iorsh, I.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2014).
[Crossref]

Jiang, H. T.

Y. Li, H. T. Jiang, W. W. Liu, J. Ran, Y. Lai, and H. Chen, “Experimental realization of subwavelength flux manipulation in anisotropic near-zero index metamaterials,” Europhys. Lett. 113(5), 57006 (2016).
[Crossref]

Jiang, M.

Jiang, X.

Kivshar, Y.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2014).
[Crossref]

Klem, J. F.

S. Campione, S. Liu, A. Benz, J. F. Klem, M. B. Sinclair, and I. Brener, “Epsilon-near-zero modes for tailored light-matter interaction,” Phys. Rev. Appl. 4(4), 044011 (2015).
[Crossref]

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
[Crossref] [PubMed]

Lai, Y.

Y. Li, H. T. Jiang, W. W. Liu, J. Ran, Y. Lai, and H. Chen, “Experimental realization of subwavelength flux manipulation in anisotropic near-zero index metamaterials,” Europhys. Lett. 113(5), 57006 (2016).
[Crossref]

Li, C.

T. Tang, C. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6(1), 30762 (2016).
[Crossref] [PubMed]

Li, Y.

I. Liberal, A. M. Mahmoud, Y. Li, B. Edwards, and N. Engheta, “Photonic doping of epsilon-near-zero media,” Science 355(6329), 1058–1062 (2017).
[Crossref] [PubMed]

Y. Li, H. T. Jiang, W. W. Liu, J. Ran, Y. Lai, and H. Chen, “Experimental realization of subwavelength flux manipulation in anisotropic near-zero index metamaterials,” Europhys. Lett. 113(5), 57006 (2016).
[Crossref]

Liberal, I.

I. Liberal, A. M. Mahmoud, Y. Li, B. Edwards, and N. Engheta, “Photonic doping of epsilon-near-zero media,” Science 355(6329), 1058–1062 (2017).
[Crossref] [PubMed]

I. Liberal and N. Engheta, “Near-zero refractive index photonics,” Nat. Photonics 11(3), 149–158 (2017).
[Crossref]

Ling, X.

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref] [PubMed]

X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
[Crossref] [PubMed]

Ling, X. H.

J. Zhang, X. X. Zhou, X. H. Ling, S. Z. Chen, H. L. Luo, and S. C. Wen, “Orbit-orbit interaction and photonic orbital Hall effect in reflection of a light beam,” Chin. Phys. B 23(6), 064215 (2014).
[Crossref]

Liu, M.

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Liu, S.

S. Campione, S. Liu, A. Benz, J. F. Klem, M. B. Sinclair, and I. Brener, “Epsilon-near-zero modes for tailored light-matter interaction,” Phys. Rev. Appl. 4(4), 044011 (2015).
[Crossref]

Liu, W. W.

Y. Li, H. T. Jiang, W. W. Liu, J. Ran, Y. Lai, and H. Chen, “Experimental realization of subwavelength flux manipulation in anisotropic near-zero index metamaterials,” Europhys. Lett. 113(5), 57006 (2016).
[Crossref]

Liu, X.

Liu, Y.

X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
[Crossref] [PubMed]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

Liu, Z.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

Löffler, W.

W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
[Crossref] [PubMed]

Lu, H.

Luo, H.

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
[Crossref] [PubMed]

Z. Xiao, H. Luo, and S. Wen, “Goos-Hanchen and Imbert-Fedorov shifts of vortex beams at air left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
[Crossref]

Luo, H. L.

J. Zhang, X. X. Zhou, X. H. Ling, S. Z. Chen, H. L. Luo, and S. C. Wen, “Orbit-orbit interaction and photonic orbital Hall effect in reflection of a light beam,” Chin. Phys. B 23(6), 064215 (2014).
[Crossref]

Luo, L.

T. Tang, C. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6(1), 30762 (2016).
[Crossref] [PubMed]

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41(17), 4032–4035 (2016).
[Crossref] [PubMed]

Luo, Y.

Mahmoud, A. M.

I. Liberal, A. M. Mahmoud, Y. Li, B. Edwards, and N. Engheta, “Photonic doping of epsilon-near-zero media,” Science 355(6329), 1058–1062 (2017).
[Crossref] [PubMed]

Merano, M.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

A. Aiello, M. Merano, and J. P. Woerdman, “Brewster cross polarization,” Opt. Lett. 34(8), 1207–1209 (2009).
[Crossref] [PubMed]

Mi, C.

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2014).
[Crossref]

Prajapati, C.

C. Prajapati, “Numerical calculation of beam shifts for higher-order Laguerre-Gaussian beams upon transmission,” Opt. Commun. 389, 290–296 (2017).
[Crossref]

Qiu, C. W.

X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
[Crossref] [PubMed]

Qiu, J.

J. Qiu, C. Ren, and Z. Zhang, “Precisely measuring the orbital angular momentum of beams via weak measurement,” Phys. Rev. A 93(6), 063841 (2016).
[Crossref]

Qiu, X.

Ran, J.

Y. Li, H. T. Jiang, W. W. Liu, J. Ran, Y. Lai, and H. Chen, “Experimental realization of subwavelength flux manipulation in anisotropic near-zero index metamaterials,” Europhys. Lett. 113(5), 57006 (2016).
[Crossref]

Ren, C.

J. Qiu, C. Ren, and Z. Zhang, “Precisely measuring the orbital angular momentum of beams via weak measurement,” Phys. Rev. A 93(6), 063841 (2016).
[Crossref]

Salandrino, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

She, W.

Sheng, L.

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref] [PubMed]

Silveirinha, M. G.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Sinclair, M. B.

S. Campione, S. Liu, A. Benz, J. F. Klem, M. B. Sinclair, and I. Brener, “Epsilon-near-zero modes for tailored light-matter interaction,” Phys. Rev. Appl. 4(4), 044011 (2015).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Stacy, A. M.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

Sun, C.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

Tang, T.

T. Tang, C. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6(1), 30762 (2016).
[Crossref] [PubMed]

Wang, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

Wen, S.

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
[Crossref] [PubMed]

Z. Xiao, H. Luo, and S. Wen, “Goos-Hanchen and Imbert-Fedorov shifts of vortex beams at air left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
[Crossref]

Wen, S. C.

J. Zhang, X. X. Zhou, X. H. Ling, S. Z. Chen, H. L. Luo, and S. C. Wen, “Orbit-orbit interaction and photonic orbital Hall effect in reflection of a light beam,” Chin. Phys. B 23(6), 064215 (2014).
[Crossref]

Woerdman, J. P.

W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
[Crossref] [PubMed]

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

A. Aiello, M. Merano, and J. P. Woerdman, “Brewster cross polarization,” Opt. Lett. 34(8), 1207–1209 (2009).
[Crossref] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Xiang, Y.

Xiao, Z.

Z. Xiao, H. Luo, and S. Wen, “Goos-Hanchen and Imbert-Fedorov shifts of vortex beams at air left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
[Crossref]

Xie, L.

Xu, Y.

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
[Crossref] [PubMed]

Yao, J.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

You, Q.

Yu, J.

Zhang, J.

Zhang, X.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

Zhang, Z.

X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41(17), 4032–4035 (2016).
[Crossref] [PubMed]

J. Qiu, C. Ren, and Z. Zhang, “Precisely measuring the orbital angular momentum of beams via weak measurement,” Phys. Rev. A 93(6), 063841 (2016).
[Crossref]

Zhong, Y.-N.

X.-T. He, Y.-N. Zhong, Y. Zhou, Z. C. Zhong, and J. W. Dong, “Dirac directional emission in anisotropic zero refractive index photonic crystals,” Sci. Rep. 5(1), 13085 (2015).
[Crossref] [PubMed]

Zhong, Z. C.

X.-T. He, Y.-N. Zhong, Y. Zhou, Z. C. Zhong, and J. W. Dong, “Dirac directional emission in anisotropic zero refractive index photonic crystals,” Sci. Rep. 5(1), 13085 (2015).
[Crossref] [PubMed]

Zhou, X.

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref] [PubMed]

X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
[Crossref] [PubMed]

Zhou, X. X.

J. Zhang, X. X. Zhou, X. H. Ling, S. Z. Chen, H. L. Luo, and S. C. Wen, “Orbit-orbit interaction and photonic orbital Hall effect in reflection of a light beam,” Chin. Phys. B 23(6), 064215 (2014).
[Crossref]

Zhou, Y.

X.-T. He, Y.-N. Zhong, Y. Zhou, Z. C. Zhong, and J. W. Dong, “Dirac directional emission in anisotropic zero refractive index photonic crystals,” Sci. Rep. 5(1), 13085 (2015).
[Crossref] [PubMed]

Zhu, W.

Zhuo, L.

Appl. Phys. Lett. (1)

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Chin. Phys. B (1)

J. Zhang, X. X. Zhou, X. H. Ling, S. Z. Chen, H. L. Luo, and S. C. Wen, “Orbit-orbit interaction and photonic orbital Hall effect in reflection of a light beam,” Chin. Phys. B 23(6), 064215 (2014).
[Crossref]

Europhys. Lett. (1)

Y. Li, H. T. Jiang, W. W. Liu, J. Ran, Y. Lai, and H. Chen, “Experimental realization of subwavelength flux manipulation in anisotropic near-zero index metamaterials,” Europhys. Lett. 113(5), 57006 (2016).
[Crossref]

J. Opt. (1)

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

Nat. Photonics (2)

I. Liberal and N. Engheta, “Near-zero refractive index photonics,” Nat. Photonics 11(3), 149–158 (2017).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2014).
[Crossref]

New J. Phys. (1)

J. B. Götte and M. R. Dennis, “Generalized shifts and weak values for polarization components of reflected light beams,” New J. Phys. 14(7), 073016 (2012).
[Crossref]

Opt. Commun. (1)

C. Prajapati, “Numerical calculation of beam shifts for higher-order Laguerre-Gaussian beams upon transmission,” Opt. Commun. 389, 290–296 (2017).
[Crossref]

Opt. Lett. (5)

Photon. Res. (2)

Phys. Rev. A (4)

J. Qiu, C. Ren, and Z. Zhang, “Precisely measuring the orbital angular momentum of beams via weak measurement,” Phys. Rev. A 93(6), 063841 (2016).
[Crossref]

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

Z. Xiao, H. Luo, and S. Wen, “Goos-Hanchen and Imbert-Fedorov shifts of vortex beams at air left-handed-material interfaces,” Phys. Rev. A 85(5), 053822 (2012).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Appl. (1)

S. Campione, S. Liu, A. Benz, J. F. Klem, M. B. Sinclair, and I. Brener, “Epsilon-near-zero modes for tailored light-matter interaction,” Phys. Rev. Appl. 4(4), 044011 (2015).
[Crossref]

Phys. Rev. B (1)

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Phys. Rev. Lett. (1)

W. Löffler, A. Aiello, and J. P. Woerdman, “Observation of orbital angular momentum sidebands due to optical reflection,” Phys. Rev. Lett. 109(11), 113602 (2012).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
[Crossref] [PubMed]

Sci. Rep. (4)

T. Tang, C. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6(1), 30762 (2016).
[Crossref] [PubMed]

X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref] [PubMed]

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
[Crossref] [PubMed]

X.-T. He, Y.-N. Zhong, Y. Zhou, Z. C. Zhong, and J. W. Dong, “Dirac directional emission in anisotropic zero refractive index photonic crystals,” Sci. Rep. 5(1), 13085 (2015).
[Crossref] [PubMed]

Science (4)

I. Liberal, A. M. Mahmoud, Y. Li, B. Edwards, and N. Engheta, “Photonic doping of epsilon-near-zero media,” Science 355(6329), 1058–1062 (2017).
[Crossref] [PubMed]

M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
[Crossref] [PubMed]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
[Crossref] [PubMed]

Other (1)

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Figures (6)

Fig. 1
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.
Fig. 2
Fig. 2 (a) The normalized m-GH shift ΔXm up changing with the incident angle θ and initial amplitude ratio β when m = 1. (b) The m-GH shift ΔXm up changing with β for m = ± 1 when θ = 5° (red), 10° (blue), 30° (green), respectively.
Fig. 3
Fig. 3 The normalized m-GH shifts ΔXm 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.
Fig. 4
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.
Fig. 5
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.
Fig. 6
Fig. 6 The Numerical verification of the theoretically predicted m-GH shifts. The normalized intensity distributions in xgzg 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.

Equations (12)

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E ˜ t = γ = p , s α γ t γ [ | ϕ ˜ 0 m ( X γ k x + Y γ k y ) / k d | ϕ ˜ 0 m ] | Γ γ . ,
t p = 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 = 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 ] ,
E t = γ = p , s α γ t γ { ϕ 0 m + θ 0 2 2 ( | m | + 1 ) [ ( | m | + 1 ) Z γ sgn [ m ] | ϕ 0 sgn [ m ] ( | m | + 1 ) | m | Z γ sgn [ m ] | ϕ 0 sgn [ m ] ( | m | 1 ) Z γ sgn [ m ] | ϕ 1 sgn [ m ] ( | m | 1 ) ] } | Γ γ ,
Δ X = γ = p , s | α γ t γ | 2 [ Re ( X γ ) + m Im ( Y γ ) ] / ( k 0 W ) ,
W = γ = p , s | α γ t γ | 2 { 1 + θ 0 2 [ | X γ | 2 + | Y γ | 2 ] / 4 } ,
Δ X m = 1 k d β cos δ [ | t p | 2 | t s | 2 ] cot θ | t p | 2 + β 2 | t s | 2 + θ 0 2 4 [ | t p ' | 2 + β 2 | t s ' | 2 + ( 1 + β 2 ) | M | 2 ] .
Δ X m , p k ± = ± m ( | t p | 2 | t s | 2 ) cot θ 2 k d | t p | 2 + θ 0 2 [ | t p ' | 2 + | M | 2 ] / 4 | t s | 2 + θ 0 2 [ | t s ' | 2 + | M | 2 ] / 4 ,
β p k ± ( θ ) = ± | t p | 2 + θ 0 2 [ | t p ' | 2 + | M | 2 ] / 4 | t s | 2 + θ 0 2 [ | t s ' | 2 + | M | 2 ] / 4 .
Δ X m , p k ± m w 0 2 | m | + 1 1 1 + | t p ' / t s cot θ | 2 .
E t = α p k d w 0 { | t s cot θ | | m | + 1 | ϕ 0 m sgn [ m ] t s cot θ [ | m | + 1 | ϕ 0 sgn [ m ] ( | m + 1 ) + | m | | ϕ 0 sgn [ m ] ( | m | 1 ) + | ϕ 1 sgn [ m ] ( | m | 1 ) ] / 2 } | V .
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 ] .

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