Three-Dimensional Measurement of the Helicity-Dependent Forces on a Mie Particle

Lulu Liu, Andrea Di Donato, Vincent Ginis, Simon Kheifets, Arman Amirzhan, and Federico Capasso

Phys. Rev. Lett. 120, 223901 – Published 31 May 2018

Abstract

Recently, it was shown that a Mie particle in an evanescent field ought to experience optical forces that depend on the helicity of the totally internally reflected beam. As yet, a direct measurement of such helicity-dependent forces has been elusive, as the widely differing force magnitudes in the three spatial dimensions place stringent demands on a measurement’s sensitivity and range. In this study, we report the simultaneous measurement of all components of this polarization-dependent optical force by using a 3D force spectroscopy technique with femtonewton sensitivity. The vector force fields are compared quantitatively with our theoretical calculations as the polarization state of the incident light is varied and show excellent agreement. By plotting the 3D motion of the Mie particle in response to the switched force field, we offer visual evidence of the effect of spin momentum on the Poynting vector of an evanescent optical field.

Received 29 October 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.223901

Physics Subject Headings (PhySH)

Interactions & forces

Colloids

Optical tweezers

Polymers & Soft Matter

Authors & Affiliations

Lulu Liu¹, Andrea Di Donato^1,2, Vincent Ginis^1,3,*, Simon Kheifets¹, Arman Amirzhan^1,4, and Federico Capasso¹

¹Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, Massachusetts 02138, USA
²Università Politecnica delle Marche, Via Brecce Bianche 60131 Ancona, Italy
³Applied Physics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
⁴Department of Materials, Imperial College London, Prince Consort Rd, London SW7 2BP, United Kingdom

^*ginis@seas.harvard.edu

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Vol. 120, Iss. 22 — 1 June 2018

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Images

Figure 1
Schematic depicting the excitation and detection setup in our experiment. (a) The excitation is provided by an evanescent wave. The continuous wave (cw) laser generating this field ( $\sim 100 mW$ , 785 nm) is blocked periodically (10–100 Hz) by a chopper. A portion ( $\sim 5 %$ ) of the modulated light is picked off to form a reference signal for lock-in position measurement. The rest passes through, in order, a linear polarizer (LP), a half-wave plate (HWP), and a quarter-wave plate (QWP), which controls the polarization of the beam. Backscattered light from the trap beam (660 nm cw) is used to determine the particle’s $x$ and $y$ displacement in volts. BAL-X and BAL-Y are two balanced detectors whose positive and negative monitors are each fed half of the backscattered light, as sectioned by two $D$ -shaped mirrors (DM). The absolute separation $z$ between the particle and the glass surface is determined by the light it scatters from a totally internally reflecting, 637 nm detection beam, maintained at around 1 mW power. In the schematic, PD1 tracks this signal in volts. (b) Detail showing the interaction of the trapped floating probe with the amplitude-modulated evanescent field, together with the coordinate system used to describe the forces acting on the particle.
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Figure 2
Measured motion of the particle in response to a periodically on-off modulated optical force. The particle, subject to an on-off optical force modulated in amplitude by a chopper set at 50 Hz, traces an ellipse in three dimensions due to the anisotropy of the mechanical response. The response is cyclical with a period matching the square-wave fundamental frequency. (a) For a single force measurement of 100 s duration, a particle’s average trajectory in 3D space, along with its projection onto each orthogonal plane is plotted. Here the HWP angle was set to 0°. (b) The projection of a particle’s measured motion in the ( $x$ , $y$ ) plane as a half-wave plate is rotated between 0° and 80°, in steps of 10°. Because of the mechanical system’s high degree of spatial symmetry in the ( $x$ , $y$ ) plane, the particle’s motion is roughly aligned with the vector direction of the in-plane optical force. The effect of the optical field’s spin momentum on the particle motion is evident here as a change in tilt of the particle’s projected trajectory in the ( $x$ , $y$ ) plane as the evanescent field’s helicity is changed. The maximum displacement measured in the $y$ direction is around 0.5 nm.
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Figure 3
Theoretical prediction of net force on a Mie particle as a function of the polarization state of an evanescent beam. (a) The polarization traces take two different closed paths, orange line and blue line on the Poincaré sphere, depending on whether the half-wave plate or quarter-wave plate is rotated, respectively. (b) The incident linearly polarized light, from the fiber, forms an angle $α = 40 °$ with the vertical (TM) axis. The half-wave plate is rotated by angle $θ$ from 0° to 180°, while the quarter-wave plate is fixed at $β = 45 °$ . (c) Alternate path where the half-wave plate is fixed at $δ = 42.5 °$ and the quarter-wave plate is rotated by angle $θ$ from 0° to 180°. It is important to note that $α$ , $β$ , and $δ$ are fixed angles with different values. $θ$ is the angle that is varied during a trace. In (b)–(c) the light is propagating from left to right ( $f$ denotes the fast axis of the wave plates). (d)–(f) Mie theory prediction for the evanescent optical force spatial components $F_{x}$ , $F_{y}$ , and $F_{z}$ , respectively, as defined in Fig. 1, with parametrized paths shown superimposed (blue for QWP and orange for HWP). The forces are normalized to the largest force (i.e., max $F_{z}$ ).
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Figure 4
Measured optical forces on two polystyrene spheres. $F_{x} / | F_{z} |$ and $F_{y} / | F_{z} |$ , the measured, normalized, optical forces, are compared against their analytical predictions (black). The modulation frequency for the lock-in detection is 50 Hz. Although the wave plate is mounted on a rotary mount, marked in 2° increments, its manual rotation introduces some additional uncertainty in the angle of rotation. (Left) The light beam’s polarization state is changed along the blue line in Fig. 3, by rotating the quarter-wave plate from 0° to 180°. The particle has a measured radius of $1.4 μ m$ , its average height is 400 nm from the chamber surface, and the beam’s angle of incidence is 63° from normal. The average measured value for $F_{z}$ is 257 fN towards the surface. The black curve showing the analytical prediction for $F_{x} / | F_{z} |$ has been scaled by a factor of 0.9 to fit our data; the bottom curve is not scaled. (Right) The light beam’s polarization state is changed along the orange line in Fig. 3, by rotating the half-wave plate from 0° to 180°. This corresponds to two full traces of the great circle on the Poincaré sphere. The particle has a measured radius of $1.3 μ m$ , its average height is 260 nm, and the beam’s angle is the same at 63°. Average $F_{z}$ is 155 fN towards the surface. The top curve is scaled by 0.86 to fit our data; the bottom curve is not scaled. (Center) Projection view shows the angular variation of the net force as the QWP (blue) and HWP (orange) are rotated. Maximum angular change is around 7° for the QWP and 9° for the HWP. Line of sight is along the $+ z$ direction.
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