Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays

Xiaolei Wang; Bowen Zhu; Yuxin Dong; Shuai Wang; Zhuqing Zhu; Fang Bo; Xiangping Li

doi:10.1364/OE.25.026844

Optics Express
Vol. 25,
Issue 22,
pp. 26844-26852
(2017)
•https://doi.org/10.1364/OE.25.026844

Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays

Xiaolei Wang, Bowen Zhu, Yuxin Dong, Shuai Wang, Zhuqing Zhu, Fang Bo, and Xiangping Li

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Xiaolei Wang, Bowen Zhu, Yuxin Dong, Shuai Wang, Zhuqing Zhu, Fang Bo, and Xiangping Li, "Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays," Opt. Express 25, 26844-26852 (2017)

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About this Article
History
- Original Manuscript: August 25, 2017
- Revised Manuscript: October 9, 2017
- Manuscript Accepted: October 9, 2017
- Published: October 18, 2017

Abstract

In this paper, the general formula for tightly focusing radially polarized beams (RPB) superposed with off-axis vortex arrays is derived based on Richard-Wolf vector diffraction theory. The off-axis vortex breaks the rotational symmetry of the energy flow along the axial direction and leads to the spatial redistribution of intensity within the focal plane. The dependence of the consequent focal intensity redistribution on the off-axis distance of vortices as well as the numerical aperture of the lens is theoretically studied. Based on this intriguing feature, generation of equilateral-polygon-like flat-top focus (EPFF) with a flat-top area on the level of sub-λ² is realized. The demonstrated method provides new opportunities for focus shaping and holds great potentials in optical manipulation and laser fabrication.

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Fig. 1 The sketch for the generation of EPFF. Without loss of generality, triangle-like flat-top focus by using three off-axis vortices is taken as an example. (a) Arrangement of three vortices (red circular) in the pupil plane. Each vortex has equal distance from the optical axis. (b) Illustration of the phase distribution of the incident beam with three off-axis vortices, off-axis vortex induced asymmetric energy flow in the axial direction, and intensity distribution of triangle-like flat-top focus at the focal plane. (c) The optical system for generating EPFF where P (ρ, ϕ) and Q (r, φ) denote points in the object space and the image space, respectively.

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Fig. 2 (a) Variation tendency of the optical intensity along the y axis in the focal plane as the increase of off-axis distance r₀ of vortices at different NA value. The data is normalized to the maximum value of the optical intensity along the y axis in the focal plane for each r₀, respectively. (b) Transverse intensities and their constituent radial and azimuthal components obtained at variant r₀ focused by an objective lens with NA = 0.2 in Fig. 2(a).

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Fig. 3 (a) Dependence of the intensity distribution along the y axis in the focal plane on the NA of the objective lens. The data are normalized to the maximum value of the intensity along the y axis in the focal plane for each NA respectively. (b) Transverse intensity patterns and their constituent radial and azimuthal components obtained by different NA when the off-axis distance r₀ is fixed at 0.8. The scale in Fig. 3(b) is different for each NA.

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Fig. 4 Normalized Poynting vector field (color density plots) and the energy flow (white lines) along the axial direction obtained by focusing a RPB superposed with three off-axis vortices under the condition of NA = 0.6 and (a) r₀ = 0.1w (b) r₀ = 0.5w and (c) r₀ = 0.9w, respectively. The red wire frame indicates the propagation section of the phase singularity in the axial direction.

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Fig. 5 Focal intensity distribution shaped by tightly focusing a RPB superposed with three off-axis vortices. (a), (b), (c) and (d) are the azimuthal, radial, longitudinal components and total intensity pattern in the x-y plane, respectively. The white line indicates the intensity profile along the x-axis and the y-axis. (e) and (f) are the total intensity pattern in the x-z plane and the y-z plane, respectively.

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Fig. 6 Arrangement of off-axis vortices in the incident beam and corresponding EPFF. The first column shows the location and arrangement of off-axis vortex arrays. The second to the fifth column corresponds to the radial, azimuthal, longitudinal components and the total intensity pattern in the x-y plane, respectively. From top to bottom, the number N of vortices used are in turn 2, 4, 5 and 6 corresponding to bar-type-like, square-like, pentagon-like and hexagon-like flat-top focus, respectively. The flat-top areas, from top to bottom, are 0.14λ², 0.33λ², 0.31λ² and 0.35λ², respectively.

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Equations (14)

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{\vec{E}}_{i n 0} (ρ, ϕ) = E_{ρ} {\hat{e}}_{ρ} = E_{0} e x p (- \frac{ρ^{2}}{w^{2}}) {\hat{e}}_{ρ}

{\vec{E}}_{i n} (ρ, ϕ) = {\vec{E}}_{i n 0} (ρ, ϕ) {(ρ e^{j ϕ} - ρ_{1} e^{j ϕ_{1}})}^{m_{1}}

\begin{array}{l} {\vec{E}}_{i n} (ρ, ϕ) = {\vec{E}}_{i n 0} (ρ, ϕ) (^{ρ e^{j ϕ} - ρ_{1} e^{j ϕ_{1}}) m_{1}} (^{ρ e^{j ϕ} - ρ_{2} e^{j ϕ_{2}}) m_{2}} ... \times (^{ρ e^{j ϕ} - ρ_{N} e^{j ϕ_{N}}) m_{N}} \\ = {\vec{E}}_{i n 0} (ρ, ϕ) \prod_{i = 1}^{N} {(ρ e^{j ϕ} - ρ_{i} e^{j ϕ_{i}})}^{m_{i}} \end{array}

{\vec{E}}_{i n} (ρ, ϕ) = {\vec{E}}_{i n 0} (ρ, ϕ) \sum_{i = 0}^{M} A_{n} ρ^{M - i} e^{j (M - i) ϕ}

M = \sum_{n = 1}^{N} m_{n}

\begin{array}{l} \vec{E} (r, φ, z) = \frac{- j k f}{2 π} \int_{0}^{α} d θ \int_{0}^{2 π} P (θ) (\begin{array}{l} - E_{i n} \cos θ \cos ϕ - φ) \\ - E_{i n} \cos θ \sin (ϕ - φ) \\ E_{i n} \sin θ \end{array}) \\ \times \exp [j k (z \cos θ + r \sin θ \cos (φ - ϕ))] \sin θ d ϕ \end{array}

\begin{array}{l} \vec{E} (r, φ, z) = (\begin{array}{l} E_{r} \\ E_{φ} \\ E_{z} \end{array}) = \frac{- j k f E_{0}}{2 π} \sum_{n = 0}^{M} \int_{0}^{α} d θ \int_{0}^{2 π} A_{n} ρ^{M - n} e^{j (M - n) ϕ} \sqrt{\cos θ} \sin θ (\begin{array}{l} - \cos θ \cos (ϕ - φ) \\ - \cos θ \sin (ϕ - φ) \\ \sin θ \end{array}) \\ \times e x p (- \frac{ρ^{2}}{w^{2}}) \exp [j k (z \cos θ + r \sin θ \cos (φ - ϕ))] d ϕ \end{array}

\begin{array}{l} E_{r} (r, φ, z) = \frac{k E_{0}}{2} \sum_{l = 0}^{M} A_{M - l} j^{l + 2} f^{l + 1} e x p (j l φ) \int_{0}^{α} {(\sin θ)}^{l + 1} {(\cos θ)}^{\frac{3}{2}} e x p (- \frac{f^{2} {(\sin θ)}^{2}}{w^{2}}) \\ \times \exp [j k z \cos θ] [J_{l + 1} (k r \sin θ) - J_{l - 1} (k r \sin θ)] d θ \end{array}

\begin{array}{l} E_{φ} (r, φ, z) = \frac{k E_{0}}{2} \sum_{l = 0}^{M} A_{M - l} j^{l + 1} f^{l + 1} e x p (j l φ) \int_{0}^{α} {(\sin θ)}^{l + 1} {(\cos θ)}^{\frac{3}{2}} e x p (- \frac{f^{2} {(\sin θ)}^{2}}{w^{2}}) \\ \times \exp [j k z \cos θ] [J_{l + 1} (k r \sin θ) + J_{l - 1} (k r \sin θ)] d θ \end{array}

\begin{array}{l} E_{z} (r, φ, z) = - k E_{0} \sum_{l = 0}^{M} A_{M - l} j^{l + 1} f^{l + 1} e x p (j l φ) \int_{0}^{α} {(\sin θ)}^{l + 2} {(\cos θ)}^{\frac{1}{2}} \\ \times e x p (- \frac{f^{2} {(\sin θ)}^{2}}{w^{2}}) \exp [j k z \cos θ] J_{l} (k r \sin θ) d θ \end{array}

H_{r} (r, φ, z) = - \frac{i}{k} (\frac{1}{r} \frac{\partial E_{z}}{\partial φ} - \frac{\partial E_{φ}}{\partial z})

H_{φ} (r, φ, z) = - \frac{i}{k} (\frac{\partial E_{r}}{\partial z} - \frac{\partial E_{z}}{\partial r})

H_{z} (r, φ, z) = - \frac{i}{k} \frac{1}{r} (\frac{\partial (ρ E_{φ})}{\partial r} - \frac{\partial E_{r}}{\partial φ})

〈 \vec{S} 〉 = \frac{c}{4 π} Re (\vec{E} \times {\vec{H}}^{*})

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Equations (14)

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