Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation

Jun Chen, Jack Ng, Shiyang Liu, and Zhifang Lin

Phys. Rev. E 80, 026607 – Published 27 August 2009

Abstract

We investigate the optical trapping of a Rayleigh particle by a linearly or radially polarized Gaussian beam. The Mie theory is applied to obtain a full solution, with the incident beam being described by the mixed dipole model, which is beyond the paraxial approximation. We then obtain approximate analytical expressions for the optical force, equilibrium position, and trap stiffness for a Rayleigh particle. At equilibrium, the displacement for the particle from the focus scales like $a^{3}$ (where $a$ is the radius) for a transparent particle, owing to scattering, whereas for absorptive particles it scales like $C + D a^{2}$ , owing to absorption. The trap stiffness is found to be proportional to $a^{3}$ , in agreement with the recent experiment. The radially polarized beam is found to be superior to the linearly polarized beam in the Rayleigh regime in terms of its ability to trap. It is found that the larger the ratio of $ε_{r} / ε_{i}$ , the closer the equilibrium to the focus, and thus higher stability.

Received 1 April 2009

DOI:https://doi.org/10.1103/PhysRevE.80.026607

Authors & Affiliations

Jun Chen^1,*, Jack Ng², Shiyang Liu¹, and Zhifang Lin¹

¹Surface Physics Laboratory, Department of Physics, Fudan University, Shanghai 200433, China
²Department of Physics, The Hong Kong University of Science and Technology, Kowloon, Hong Kong

^*Author to whom correspondence should be addressed; 071019010@fudan.edu.cn

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Vol. 80, Iss. 2 — August 2009

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Images

Figure 1
(Color online) Equilibrium position as a function of complex permittivity for a sphere illuminated by a Gaussian beam. The waist radius $w_{0} = λ$ . The size parameter $ρ = 0.2$ for (a) and (b), and $ρ = 0.3$ for (c) and (d). (a) and (c) are for the linearly polarized case (x polarized), (b) and (d) are for the radially polarized case.Reuse & Permissions
Figure 2
(Color online) Axial optical force versus $z / k w_{0}^{2}$ for a polystyrene sphere in water $[ε = {(1.592 / 1.332)}^{2}]$ . The dashed black curves are calculated by the accurate Mie theory, the solid green curves are calculated by our analytical approach, and the dotted blue curves are calculated by the approach of Harada. The wavelength of Gaussian beam $λ = 514.5 nm$ and beam power $P = 100 mW$ . (a) $w_{0} = 2.2 μ m$ , $ρ = 0.2$ and (b) $w_{0} = 2.2 μ m$ , $ρ = 1.6$ , (c) $w_{0} = 0.25 μ m$ , $ρ = 0.4$ and (d) $w_{0} = 0.5 μ m$ , $ρ = 1.6$ .Reuse & Permissions
Figure 3
(Color online) Comparison for $z_{e q u} / λ$ versus $ρ$ between simplified analytical results and the accurate Mie theory. A spherical nanoparticle in air is illuminated either by a radially polarized or a linearly polarized Gaussian beam with the wavelength $λ = 1064 nm$ and the waist radius $w_{0} = λ$ . Both the transparent particle with $ε = 5$ and the absorptive particle with $ε = 5 + 0.05 i$ are considered. In the figure legend, $Mie = Mie$ Theory, $Ana = Analytical$ , $Abs = Absorption$ , $Tra = Transparent$ , $Lin = Linear$ Polarization, $Rad = Radial$ Polarization.Reuse & Permissions
Figure 4
(Color online) A spherical nanoparticle in air with $ρ = 0.5$ , illuminated by a radially polarized or a linearly polarized Gaussian beam with the wavelength $λ = 1064 nm$ and waist radius $w_{0} = λ$ . Simplified analytical calculations for both the transparent particle and the absorptive particle $(ε_{i} = 0.01)$ are considered. (a) Ratio $R$ versus $ε_{r}$ , (b) and (c) are $Q_{t o t} [= F_{t o t} / (n P / c)]$ versus $z / λ$ for a linearly and a radially polarized Gaussian beam, respectively, with $ε_{r} = 5$ , $n$ the refractive index of the surrounding medium, $P$ the beam power, and $c$ the light velocity in vacuum. In the figure legend, $Ana = Analytical$ , $Abs = Absorption$ , $Tra = Transparent$ , $Lin = Linear$ Polarization, $Rad = Radial$ Polarization.Reuse & Permissions
Figure 5
(Color online) A spherical nanoparticle in air is illuminated by either a radially polarized or a linearly polarized Gaussian beam with the wavelength $λ = 1064 nm$ and waist radius $w_{0} = λ$ . Both the transparent particle $(ε = 1.5)$ and the absorptive particle $(ε = 1.5 + 0.01 i)$ are considered. (a) Ratio $R$ versus $ε_{i}$ with $ρ = 0.5$ , (b) Ratio $R$ versus $ρ$ and (c) Ratio $R$ versus $z / λ$ with $ρ = 0.5$ . In the figure legend, $Ana = Analytical$ , $Abs = Absorption$ , $Tra = Transparent$ , $Lin = Linear$ Polarization, $Rad = Radial$ Polarization.Reuse & Permissions

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