Theory of optical-tweezers forces near a plane interface

R. S. Dutra, P. A. Maia Neto, H. M. Nussenzveig, and H. Flyvbjerg

Phys. Rev. A 94, 053848 – Published 28 November 2016

Abstract

Optical-tweezers experiments in molecular and cell biology often take place near the surface of the microscope slide that defines the bottom of the sample chamber. There, as elsewhere, force measurements require force-calibrated tweezers. In bulk, one can calculate the tweezers force from first principles, as recently demonstrated. Near the surface of the microscope slide, this absolute calibration method fails because it does not account for reverberations from the slide of the laser beam scattered by the trapped microsphere. Nor does it account for evanescent waves arising from total internal reflection of wide-angle components of the strongly focused beam. In the present work we account for both of these phenomena. We employ Weyl's angular spectrum representation of spherical waves in terms of real and complex rays and derive a fast-converging recursive series of multiple reflections that describes the reverberations, including also evanescent waves. Numerical simulations for typical setup parameters evaluate these effects on the optical force and trap stiffness, with emphasis on axial trapping. Results are in good agreement with available experimental data. Thus, absolute calibration now applies to all situations encountered in practice.

Received 29 August 2016

DOI:https://doi.org/10.1103/PhysRevA.94.053848

Physics Subject Headings (PhySH)

Classical optics Lasers Light propagation, transmission & absorption Scattering theory

Atomic, Molecular & Optical

Authors & Affiliations

R. S. Dutra^1,2, P. A. Maia Neto^3,4, H. M. Nussenzveig^3,4, and H. Flyvbjerg²

¹LISComp-IFRJ, Instituto Federal de Educação, Ciência e Tecnologia, Rua Sebastião de Lacerda, Paracambi, RJ, 26600-000, Brazil
²Department of Micro-and Nanotechnology, Technical University of Denmark, DK-2800 Lyngby, Denmark
³LPO-COPEA, Instituto de Ciências Biomédicas, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21941-590, Brazil
⁴Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ, 21941-972, Brazil

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Issue

Vol. 94, Iss. 5 — November 2016

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Images

Figure 1
Microsphere immersed in water, with center C distant $L_{C}$ from glass slide. $PF =$ paraxial focus of incident laser beam. $E_{s}^{(0)} =$ zero-order spherical wave; $δ E_{in}^{(1)} (α, β) =$ first-order of iteration for component of incident angular spectrum in $(α, β)$ direction.
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Figure 2
Dimensionless axial force efficiency $Q_{z}$ for a microsphere with radius $a = 1.0 μ m$ as a function of the height $L_{C} \geq a$ of the center of the microsphere over the surface of the glass slide. Results are shown for three different numerical apertures ( $NA$ ) of the objective, all for the same paraxial focal position, fixed at height $L = 2.0 μ m$ over the surface of the glass slide, as indicated by the vertical dotted line.
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Figure 3
Dimensionless axial force efficiency $Q_{z}$ as a function of the microsphere height $L_{C}$ for four different values of the microsphere radius. We take $NA = 1.4$ and the paraxial focal position at $L = 2 a$ above the glass slide. The vertical dashed lines indicate the position $L_{C} = a$ of the center of the microsphere when the microsphere touches the surface of the glass slide. The solid line represents the case of a typical astigmatic incident beam (see text), for the radius $a = 1.5 μ m$ .
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Figure 4
Axial trap stiffness, normalized by the laser power in the sample chamber, as a function of the objective upward displacement $d$ , for a microsphere of $0.5 μ m$ radius and an objective with numerical aperture $NA = 1.25 .$ The value $d = 0$ corresponds to the configuration where the microsphere in equilibrium is just touching the glass slide. Solid line: MDSA+ theory. Circles: MDSA+ theory corrected for the effect of reverberation. The inset shows the relative correction of the axial equilibrium position, as a function of the objective displacement.
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Figure 5
Axial trap stiffness, normalized by the laser power in the sample chamber, as a function of objective displacement $d$ , for a microsphere of $0.5 μ m$ radius and an objective with numerical aperture $NA = 1.4$ . Solid line: MDSA+ theory taking into account the evanescent waves. Dashed line: MDSA+ theory cutting off the evanescent waves, for an effective numerical aperture $NA = 1.332$ . Circles: MDSA+ theory taking into account the evanescent waves as well as the reverberation from the glass surface. Triangles: MDSA+ theory without evanescent waves taking into account the reverberation from the glass surface. Inset: dimensionless transverse force efficiency as a function of radial position, in units of microsphere radius, for an objective displacement $d = 0$ . Solid line: MDSA+ theory taking into account the evanescent waves. Circles: MDSA+ theory taking into account the evanescent waves and the reverberation from the glass surface.
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