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Photonics Research
Vol. 5, Issue 2, 64 (2017)

Polarization evolution of vector beams generated by q-plates

Weixing Shu^1、4、*, Xiaohui Ling^2、3, Xiquan Fu¹, Yachao Liu², Yougang Ke², and Hailu Luo^2、5、*

Author Affiliations

¹College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, China

²Laboratory for Spin Photonics, School of Physics and Electronics, Hunan University, Changsha 410082, China

³College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang 421002, China

⁴e-mail: wxshu@hnu.edu.cn.

⁵e-mail: hailuluo@hnu.edu.cn.

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DOI: 10.1364/PRJ.5.000064 Cite this Article Set citation alerts

Weixing Shu, Xiaohui Ling, Xiquan Fu, Yachao Liu, Yougang Ke, Hailu Luo. Polarization evolution of vector beams generated by q-plates[J]. Photonics Research, 2017, 5(2): 64 Copy Citation Text

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Abstract

The polarization evolution of vector beams (VBs) generated by

q

-plates is investigated theoretically and experimentally. An analytical model is developed for the VB created by a general quarter-wave

q

-plate based on vector diffraction theory. It is found that the polarization distribution of VBs varies with position and the value

q

. In particular, for the incidence of circular polarization, the exit vector vortex beam has polarization states that cover the whole surface of the Poincaré sphere, thereby constituting a full Poincaré beam. For the incidence of linear polarization, the VB is not cylindrical but specularly symmetric, and exhibits an azimuthal spin splitting. These results are in sharp contrast with those derived by the commonly used model, i.e., regarding the incident light as a plane wave. By implementing

q

-plates with dielectric metasurfaces, further experiments validate the theoretical results.

Keywords

Berry''s phase Diffraction theory Form birefringence Metamaterials Optical vortices Polarization

E_{i} (x, y, 0) = E_{0} (x, y) (\cos ϑ e^{- i ϕ} | + ⟩ + \sin ϑ e^{i ϕ} | - ⟩),

(1)

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T = [\begin{matrix} \cos \frac{δ}{2} - i \sin \frac{δ}{2} \cos 2 α & - i \sin \frac{δ}{2} \sin 2 α \\ - i \sin \frac{δ}{2} \sin 2 α & \cos \frac{δ}{2} + i \sin \frac{δ}{2} \cos 2 α \end{matrix}] .

(2)

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T_{1 / 4} = \frac{1}{\sqrt{2}} [\begin{matrix} 1 - i \cos 2 α & - i \sin 2 α \\ - i \sin 2 α & 1 + \cos 2 α \end{matrix}] = \frac{1}{\sqrt{2}} (I + T_{1 / 2}),

(3)

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T_{1 / 2} = - i [\begin{matrix} \cos 2 α & \sin 2 α \\ \sin 2 α & - \cos 2 α \end{matrix}] .

(4)

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α = q θ + α_{0},

(5)

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E (x, y, 0) = \frac{E_{0}}{\sqrt{2}} [\cos ϑ e^{- i ϕ} | + ⟩ + \sin ϑ e^{i ϕ} | - ⟩] - i \frac{E_{0}}{\sqrt{2}} [\sin ϑ e^{- i (2 α - ϕ)} | + ⟩ + \cos ϑ e^{i (2 α - ϕ)} | - ⟩] .

(6)

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E (r, z) = \frac{e^{i k z}}{i λ z} \int_{0}^{\infty} \int_{0}^{2 π} ρ d ρ d φ E (ρ, 0) \times \exp [i k \frac{ρ^{2}}{2 z} - i \frac{k r ρ}{z} \cos (φ - θ)],

(7)

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E (r, z) = \frac{1}{\sqrt{2}} [E_{i} (r, z) - i E_{1 / 2} (r, z)] .

(8)

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E_{i} (r, z) = A_{1} [\cos ϑ e^{- i ϕ} | + ⟩ + \sin ϑ e^{i ϕ} | - ⟩]

(9)

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E_{1 / 2} (r, z) = A_{2} [\sin ϑ e^{- i (2 α - ϕ)} | + ⟩ + \cos ϑ e^{i (2 α - ϕ)} | - ⟩] .

(10)

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A_{1} = e^{i k z - \frac{k r^{2}}{k w_{0}^{2} + i 2 z}} \frac{k w_{0}^{2}}{k w_{0}^{2} + i 2 z},

(11)

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A_{2} = e^{i k z + i k \frac{r^{2}}{2 z}} \frac{k w_{0}^{2}}{k w_{0}^{2} + i 2 z} {(\frac{- i k^{2} r^{2} w_{0}^{2}}{2 k w_{0}^{2} z + i 4 z^{2}})}^{| q |} \times \frac{2^{- 2 | q |} \sqrt{π}}{Γ (1 / 2 + | q |)} {}_{1}{F_{1}} (1 + | q |, 1 + 2 | q |, - \frac{i k^{2} r^{2} w_{0}^{2}}{2 k w_{0}^{2} z + i 4 z^{2}}) .

(12)

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E_{\pm} (x, y, z) = \frac{1}{\sqrt{2}} [A_{1} | \pm ⟩ - i A_{2} e^{\pm i 2 α} | \mp ⟩]

(13)

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= A e^{\pm i β} [\cos χ e^{\mp i β} | + ⟩ + \sin χ e^{\pm i β} | - ⟩],

(14)

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E_{p \pm} (x, y, 0) = \frac{E_{0}}{\sqrt{2}} [| \pm ⟩ - i e^{\pm i 2 α} | \mp ⟩] = E_{0} e^{\pm i (α \mp π / 4)} [\begin{matrix} \cos (α \mp π / 4) \\ \sin (α \mp π / 4) \end{matrix}] .

(15)

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E (r, z) = \frac{1}{\sqrt{2}} [\begin{matrix} A_{1} \cos ϕ - i A_{2} \cos (2 α - ϕ) \\ A_{1} \sin ϕ - i A_{2} \sin (2 α - ϕ) \end{matrix}] .

(16)

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E_{p} (r, z) = \frac{E_{0}}{\sqrt{2}} [\begin{matrix} \cos ϕ - i \cos (2 α - ϕ) \\ \sin ϕ - i \sin (2 α - ϕ) \end{matrix}] .

(17)

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Weixing Shu, Xiaohui Ling, Xiquan Fu, Yachao Liu, Yougang Ke, Hailu Luo. Polarization evolution of vector beams generated by q-plates[J]. Photonics Research, 2017, 5(2): 64

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