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Advanced Photonics Nexus
Vol. 2, Issue 2, 026001 (2023)

Nondiffractive three-dimensional polarization features of optical vortex beams

Andrei Afanasev^1、*, Jack J. Kingsley-Smith², Francisco J. Rodríguez-Fortuño², and Anatoly V. Zayats²

Author Affiliations

¹The George Washington University, Department of Physics, Washington, DC, United States

²King’s College London, Department of Physics and London Centre for Nanotechnology, London, United Kingdom

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DOI: 10.1117/1.APN.2.2.026001 Cite this Article Set citation alerts

Andrei Afanasev, Jack J. Kingsley-Smith, Francisco J. Rodríguez-Fortuño, Anatoly V. Zayats. Nondiffractive three-dimensional polarization features of optical vortex beams[J]. Advanced Photonics Nexus, 2023, 2(2): 026001 Copy Citation Text

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Abstract

Vector optical vortices exhibit complex polarization patterns due to the interplay between spin and orbital angular momenta. Here we demonstrate, both analytically and with simulations, that certain polarization features of optical vortex beams maintain constant transverse spatial dimensions independently of beam divergence due to diffraction. These polarization features appear in the vicinity of the phase singularity and are associated with the presence of longitudinal electric fields. The predicted effect may prove important in metrology and high-resolution imaging applications.

Keywords

diffraction optical angular momentum polarization

E_{⊥} (r) = η_{⊥} A_{⊥} \frac{w_{0}}{w (z)} {[\frac{ρ}{w (z)}]}^{| l |} e^{- \frac{ρ^{2}}{w^{2} (z)}} e^{i [l ϕ + k z + k \frac{ρ^{2}}{2 R (z)} - (l + 1) ψ (z)]},

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\nabla_{⊥} \cdot E_{⊥} = - i k E_{z} .

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\frac{E_{z}}{E_{x}} = \frac{i ρ \cos ϕ}{z_{R} + \frac{z^{2}}{z_{R}}} .

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R_{LT} \equiv \frac{| E_{z} |}{| E_{⊥} |} = \frac{ρ}{z_{R} + \frac{z^{2}}{z_{R}}},

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R_{LT} (ζ) = {\begin{cases} \sqrt{2} | ζ |, & circular (σ \cdot l < 0) polarization \\ 0, & circular (σ \cdot l > 0) polarization \\ | ζ |, & linear polarization \\ 2 | ζ |, & radial polarization \\ 0, & azimuthal polarization \end{cases},

(5)

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ρ_{m n} = \frac{1}{3} {I + \frac{3}{2} \sum_{i = x, y, z} p_{i} P_{i} + \sum_{i, j = x, y, z} p_{i j} P_{i j}}_{m n},

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{| E |}^{2} p_{n} = i \sum_{j k} ϵ_{n j k} E_{j} E_{k}^{*}, {| E |}^{2} p_{n k} = - \frac{3}{2} (E_{n} E_{k}^{*} + E_{k} E_{n}^{*} - \frac{2 {| E |}^{2}}{3} δ_{n k}),

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{| E |}^{2} p_{x} = \frac{1}{\sqrt{2}} [(E^{-} + E^{+}) E_{z}^{*} + E_{z} (E^{-} + E^{+})^{*}], {| E |}^{2} p_{y} = \frac{i}{\sqrt{2}} [E_{z} (E^{-} - E^{+})^{*} - (E^{-} - E^{+}) E_{z}^{*}], {| E |}^{2} p_{z} = {| E^{+} |}^{2} - {| E^{-} |}^{2}, {| E |}^{2} (p_{x x} - p_{y y}) = 3 (E^{+} E^{- *} + E^{-} E^{+ *}), {| E |}^{2} p_{z z} = {| E^{+} |}^{2} + {| E^{-} |}^{2} - 2 {| E_{z} |}^{2}, {| E |}^{2} p_{x y} = i \frac{3}{2} (E^{+} E^{- *} - E^{-} E^{+ *}), {| E |}^{2} p_{x z} = \frac{3}{2 \sqrt{2}} (E^{+} E_{z}^{*} + E_{z} E^{+ *} - E^{-} E_{z} * - E_{z} E^{- *}), | E |^{2} p_{y z} = i \frac{3}{2 \sqrt{2}} (E^{+} E_{z}^{*} - E_{z} E^{+ *} + E^{-} E_{z}^{*} - E_{z} E^{- *}) .

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p_{z} \to \frac{S_{3}}{S_{0}}, p_{x x} - p_{y y} \to - \frac{3 S_{1}}{S_{0}}, p_{x y} = - \frac{3 S_{2}}{2 S_{0}},

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p_{z z} (ζ) = {\begin{cases} \frac{1 - 4 ζ^{2}}{1 + 2 ζ^{2}}, & circular (σ \cdot l < 0) polarization \\ 1, & circular (σ \cdot l > 0) polarization \\ \frac{1 - 2 ζ^{2}}{1 + ζ^{2}}, & linear polarization \\ \frac{1 - 8 ζ^{2}}{1 + 4 ζ^{2}}, & radial polarization \\ 1 & azimuthal polarization \end{cases} .

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E (r) = \iint (A_{p} {\hat{e}}_{p} + A_{s} {\hat{e}}_{s}) e^{i (k \cdot r)} d k_{x} d k_{y},

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S_{0} = E_{x} E_{x}^{*} + E_{y} E_{y}^{*}, S_{1} = E_{x} E_{x}^{*} - E_{y} E_{y}^{*}, S_{2} = E_{x} E_{y}^{*} + E_{y} E_{x}^{*}, S_{3} = i (E_{x} E_{y}^{*} - E_{y} E_{x}^{*}) .

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\frac{S_{1}^{(x z)}}{S_{0}} = \frac{p_{z z} - p_{x x}}{3}, \frac{S_{2}^{(x z)}}{S_{0}} = - \frac{2}{3} p_{x z}, \frac{S_{3}^{(x z)}}{S_{0}} = - p_{y} .

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E_{⊥} (r) = η_{⊥} A_{⊥} \frac{w_{0}}{w (z)} {[\frac{ρ}{w (z)}]}^{| l |} L_{m}^{l} [\frac{2 ρ^{2}}{w^{2} (z)}] e^{- \frac{ρ^{2}}{w^{2} (z)}} e^{i [l ϕ + k z + k \frac{ρ^{2}}{2 R (z)} - (l + 2 m + 1) ψ (z)]},

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E_{⊥} (k_{x}, k_{y}) = \frac{1}{4 π^{2}} \iint E_{⊥} (x, y) e^{- i (k_{x} x + k_{y} y)} d x d y .

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E_{⊥} (k_{x}, k_{y}) = i η_{⊥} A_{⊥} \frac{π w_{0}^{3}}{2} e^{\frac{- w_{0}^{2} (k_{x}^{2} + k_{y}^{2})}{4}} (k_{x} + i k_{y}) .

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E_{⊥} = A_{p} {\hat{e}}_{p} + A_{s} {\hat{e}}_{s} - E_{z} \hat{z} .

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E_{⊥} = A_{p} [{\hat{e}}_{p} - ({\hat{e}}_{p} \cdot \hat{z}) \hat{z}] + A_{s} {\hat{e}}_{s} .

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E_{⊥} \cdot {\hat{e}}_{p} = A_{p} (1 - {({\hat{e}}_{p} \cdot \hat{z})}^{2}), E_{⊥} \cdot {\hat{e}}_{s} = A_{s} .

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A_{p} = (E_{⊥} \cdot {\hat{e}}_{p}) {(\frac{k}{k_{z}})}^{2}, A_{s} = E_{⊥} \cdot {\hat{e}}_{s} .

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A_{p} = i A_{⊥} \frac{w_{0}^{3}}{8 π} e^{\frac{- w_{0}^{2} (k_{x}^{2} + k_{y}^{2})}{4}} (k_{x} + i k_{y}) k \frac{k_{x} η_{x} + k_{y} η_{y}}{k_{z} \sqrt{k_{x}^{2} + k_{y}^{2}}}, A_{s} = i A_{⊥} \frac{w_{0}^{3}}{8 π} e^{\frac{- w_{0}^{2} (k_{x}^{2} + k_{y}^{2})}{4}} (k_{x} + i k_{y}) \frac{- k_{y} η_{x} + k_{x} η_{y}}{\sqrt{k_{x}^{2} + k_{y}^{2}}},

(21)

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Andrei Afanasev, Jack J. Kingsley-Smith, Francisco J. Rodríguez-Fortuño, Anatoly V. Zayats. Nondiffractive three-dimensional polarization features of optical vortex beams[J]. Advanced Photonics Nexus, 2023, 2(2): 026001

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