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Journals >Photonics Research >Volume 9 >Issue 8 >Page 1598 > Article

Photonics Research
Vol. 9, Issue 8, 1598 (2021)

Formation of a three-dimensional bottle beam via an engineered microsphere

Yan Zhou and Minghui Hong^*

Author Affiliations

Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore

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DOI: 10.1364/PRJ.430514 Cite this Article Set citation alerts

Yan Zhou, Minghui Hong. Formation of a three-dimensional bottle beam via an engineered microsphere[J]. Photonics Research, 2021, 9(8): 1598 Copy Citation Text

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Abstract

In this work, we propose a novel approach to produce three-dimensional (3D) optical trapping with sub-wavelength size through an engineered microsphere, under linear polarization states of an incident light. The engineered microsphere is designed to contain the segmented regions of diffractive patterns and made by focused ion beam fabrication. We simulate and experimentally characterize the focus performance of the engineered microsphere. The emitted light field from the exit surface of the engineered microsphere forms a pair of axially arranged focused beams, and they are connected with a continuous optical field embracing a 3D optical null at the center, forming the so-called optical bottle beam. Experimental results and numerical simulation are in good agreement. Such micro-optics can be used for precise and localized optical trapping.

φ = - \frac{n^{2} r^{4}}{8 f^{3} {(n - 1)}^{2}},

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S_{I} = \frac{n}{4 {(n - 1)}^{2}} - \frac{1}{4},

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A_{pl} = \exp (i k z),

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A_{G} = \frac{w_{0}}{w (z)} \exp [- \frac{ρ^{2}}{w^{2} (z)}] \times \exp {i [k z + \frac{k ρ^{2}}{2 R (z)} - \arctan \frac{z - z_{0}}{z_{R}}]},

(4)

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I (u, v) = \frac{8 π α^{4} P}{λ^{2} f^{2} w_{0}^{2}} {| \int_{0}^{1} \exp (- \frac{ζ^{2}}{w^{2}}) \exp [- i (k φ - \frac{u ζ^{2}}{2})] J_{0} (v ζ) ζ d ζ |}^{2},

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r_{m} = \sqrt{m f_{d} λ + {(m λ / 2)}^{2}}, m = 1, 2, 3, \dots, 2 N + 1,

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E_{x}^{rid} (ρ, η, z) = - \frac{z}{2 π} \sum_{m = 0}^{N} \int_{r_{2 m}}^{r_{2 m + 1}} \int_{0}^{2 π} \int_{- π / 2}^{π / 2} E_{0 x}^{rid} (r, φ; σ) \frac{i k l - 1}{l^{3}} \exp (i k l) r d r d φ d σ, E_{y}^{rid} (ρ, η, z) = - \frac{z}{2 π} \sum_{m = 0}^{N} \int_{r_{2 m}}^{r_{2 m + 1}} \int_{0}^{2 π} \int_{- π / 2}^{π / 2} E_{0 y}^{rid} (r, φ; σ) \frac{i k l - 1}{l^{3}} \exp (i k l) r d r d φ d σ, E_{z}^{rid} (ρ, η, z) = \frac{z}{2 π} \sum_{m = 0}^{N} \int_{r_{2 m}}^{r_{2 m + 1}} \int_{0}^{2 π} \int_{- π / 2}^{π / 2} [(ρ \cos η - r \cos φ) E_{0 x}^{rid} (r, φ; σ) (ρ \sin η - r \sin φ) E_{0 y}^{rid} (r, φ; σ)] \frac{i k l - 1}{l^{3}} \exp (i k l) r d r d φ d σ,

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E_{x, n}^{gro} (ρ, η, z) = - \frac{z}{2 π} \sum_{m = 1}^{N} \int_{r_{2 m - 1}}^{r_{2 m}} \int_{0}^{2 π} \int_{σ_{n, 1}}^{σ_{n, 2}} E_{0 x, n}^{gro} (r^{'}, φ; σ) \frac{i k l_{n} - 1}{l^{3}} \exp (i k l_{n}) r^{'} ξ_{n} d r d φ d σ, E_{y, n}^{gro} (ρ, η, z) = - \frac{z}{2 π} \sum_{m = 1}^{N} \int_{r_{2 m - 1}}^{r_{2 m}} \int_{0}^{2 π} \int_{σ_{n, 1}}^{σ_{n, 2}} E_{0 y, n}^{gro} (r^{'}, φ; σ) \frac{i k l_{n} - 1}{l^{3}} \exp (i k l_{n}) r^{'} ξ_{n} d r d φ d σ, E_{z, n}^{gro} (ρ, η, z) = \frac{z}{2 π} \sum_{m = 1}^{N} \int_{r_{2 m - 1}}^{r_{2 m}} \int_{0}^{2 π} \int_{σ_{n, 1}}^{σ_{n, 2}} [(ρ \cos η - r^{'} \cos φ) E_{0 x, n}^{gro} (r, φ; σ) + (ρ \sin η - r^{'} \sin φ) E_{0 y, n}^{gro} (r, φ; σ)] \frac{i k l_{n} - 1}{l^{3}} \exp (i k l_{n}) r^{'} ξ_{n} d r d φ d σ,

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Φ (x, y) = {\begin{matrix} 2 π - K x, \begin{matrix} y > 0 \end{matrix} \\ π + K x, \begin{matrix} y \leq 0 \end{matrix} \end{matrix},

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Φ (x, y) = {\begin{matrix} 2 π - K_{A} x, \begin{matrix} x > 0, y > 0 \end{matrix} \\ π + K_{B} x, \begin{matrix} x > 0, y \leq 0 \end{matrix} \\ π + K_{A} x, \begin{matrix} x \leq 0, y \leq 0 \end{matrix} \\ 2 π - K_{B} x, \begin{matrix} x \leq 0, y > 0 \end{matrix} \end{matrix},

(10)

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Yan Zhou, Minghui Hong. Formation of a three-dimensional bottle beam via an engineered microsphere[J]. Photonics Research, 2021, 9(8): 1598

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