Goos-H\"anchen and Imbert-Fedorov shifts of vortex beams at air--left-handed-material interfaces

Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces

Zhicheng Xiao, Hailu Luo, and Shuangchun Wen

Phys. Rev. A 85, 053822 – Published 18 May 2012

Abstract

In this paper, we present a systematic study of beam shifts and angular momenta of paraxial vortex beams at air–left-handed-material (LHM) interfaces. It is shown that, compared to their counterparts at air–right-handed-material (RHM) interfaces, the spatial Goos-Hänchen (GH) and Imbert-Fedorov (IF) shifts remain the same, while the angular GH and IF shifts are reversed at air-LHM interfaces. The spatial and angular shifts of paraxial vortex beams have their respective origins in transverse angular momenta and transverse linear momenta. The spatial GH and IF shifts remain unreversed as a result of both reversions of transverse angular momenta and $z$ -component linear momentum, while the angular GH and IF shifts are reversed because the $z$ -component linear momentum is reversed and the transverse linear momenta are unreversed at air-LHM interfaces. In addition, we perform a quantitative analysis on spin-orbit angular momentum conversion and orbit-orbit angular momentum conversion, which further helps us understand the essence of vortex beam shifts at air-LHM interfaces and their fundamental distinctions from those at air-RHM interfaces.

Received 14 February 2012

DOI:https://doi.org/10.1103/PhysRevA.85.053822

Authors & Affiliations

Zhicheng Xiao, Hailu Luo^*, and Shuangchun Wen^†

Key Laboratory for Micro/Nano Optoelectronic Devices of Ministry of Education and School of Information Science and Engineering, Hunan University, Changsha 410082, China

^*hailuluo@hnu.edu.cn
^†scwen@hnu.edu.cn

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Vol. 85, Iss. 5 — May 2012

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Figure 1
Geometry of reflection and transmission. The incident plane of the main Fourier component is $x o z$ [marked with green (light gray) lines]. For an arbitrary plane wave, the incident plane is $X O Z$ [marked with blue (dark gray) lines].Reuse & Permissions
Figure 2
The helical wave fronts of incident, reflected, and transmitted beams. Incident, reflected, and transmitted vortex beams propagate along the $z_{i}, z_{r}$ , and $z_{t}$ axes, respectively. We choose $l = 1$ for the incident vortex beam. The incident, reflected, and transmitted vortex beams have counterclockwise, clockwise, and clockwise helical wave fronts, respectively.Reuse & Permissions
Figure 3
The GH and IF shifts in TIR. The parameters are incident angle $θ_{i} = π / 4$ , refractive index of LHM $n_{L} = - 1.515$ , refractive index of RHM $n_{R} = 1.515$ , vortex charge $l = 1$ , propagation distance $z = z_{R}$ , and beam waist $w_{0} = 20 λ$ , where $λ$ is the wavelength in vacuum. To make the shifts more noticeable, we amplify them by 20 times. In (a) and (b), the incident beam is left circularly polarized. The actual beam shifts of (a) are $〈 x_{r}^{L} 〉 = 1.129 λ$ and $〈 y_{r}^{L} 〉 = - 0.282 λ$ . The actual beam shifts of (b) are $〈 x_{r}^{R} 〉 = - 1.129 λ$ and $〈 y_{r}^{R} 〉 = - 0.282 λ$ . In (c) and (d), the incident beam is mixed linearly polarized with $χ = 1$ . The actual beam shifts of (c) are $〈 x_{r}^{L} 〉 = 1.129 λ$ and $〈 y_{r}^{L} 〉 = 0.101 λ$ . The actual beam shifts of (d) are $〈 x_{r}^{R} 〉 = - 1.129 λ$ and $〈 y_{r}^{R} 〉 = - 0.101 λ$ .Reuse & Permissions
Figure 4
The spatial GH and IF shifts of transmitted vortex beams at air-LHM and air-RHM interfaces. The parameters are incident angle $θ_{i} = π / 4$ , refractive index of LHM $n_{L} = - 1.515$ , refractive index of RHM $n_{R} = 1.515$ , polarization state $σ = 1 / 2$ , $χ = 1 / 2$ , propagation distance $z = z_{R}$ , and beam waist $w_{0} = 20 λ$ . To make the shifts more noticeable, we amplify them by a factor of 200. In (a) and (b), the vortex charge is $l = 1$ . The actual beam shifts of (a) and (b) are $Δ x_{t}^{L} = 0.006 λ$ and $Δ y_{t}^{L} = 0.085 λ$ . In (c) and (d), the vortex charge is $l = - 1$ . The actual beam shifts of (c) and (d) are $Δ x_{t}^{L} = - 0.006 λ$ and $Δ y_{t}^{L} = - 0.033 λ$ .Reuse & Permissions
Figure 5
The spatial GH and IF shifts of transmitted vortex beams at air-LHM and air-RHM interfaces. The parameters are the refractive index of LHM $n_{L} = - 1.515$ , the refractive index of RHM $n_{R} = 1.515$ , polarization state $σ = 1 / 2$ , and $χ = 1 / 2$ . The spatial shifts are presented on the scale of $λ$ .Reuse & Permissions
Figure 6
The angular GH and IF shifts of transmitted vortex beams at air-LHM and air-RHM interfaces. The parameters are the refractive index of LHM $n_{L} = - 1.515$ , the refractive index of RHM $n_{R} = 1.515$ , polarization state $σ = 1 / 2$ , and $χ = 1 / 2$ . The angular shifts are presented on the scale of $θ_{0}^{2}$ .Reuse & Permissions

Physical Review A

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Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces

Zhicheng Xiao, Hailu Luo, and Shuangchun Wen

Phys. Rev. A 85, 053822 – Published 18 May 2012

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Physical Review A

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Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces

Zhicheng Xiao, Hailu Luo, and Shuangchun Wen

Phys. Rev. A 85, 053822 – Published 18 May 2012

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