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WHY DO ELLIPSOIDAL PARTICLES OSCILLATE IN FOCUSED GAUSSIAN BEAMS?Jean-Christophe LOUDET1,*, Bernard POULIGNY 1 and Besira MIHIRETIE 1, 2

1 Université de Bordeaux, CNRS Centre de recherche Paul Pascal, Pessac, 33600, France2 Department of Physics, University of Gothenburg, Sweden

*Corresponding author: loudet@crpp-bordeaux.cnrs.fr

AbstractRecently dielectric particles of ellipsoidal shapes have

been observed to undergo sustained oscillations driven bylaser radiation pressure forces and torques. We show thatthe observed particle dynamics can be understood througha simple ray-optics model, including the influence of beamdivergence.

1 IntroductionSpherical dielectric particles are commonly

manipulated with laser beams. Within certain limits (sizeand refractive index) a little sphere can be locked aroundthe focus of a large aperture beam, as in optical tweezers[1]. An alternate – and historically earlier - configuration isthat of optical levitation, whereby a vertical moderatelyfocused beam is used to lift the particle against its ownweight [2]. The particle may be stopped near the beam-waist: the equilibrium so obtained is not strictly stable, butmay be easily maintained by tuning the laser power.

We recently carried out levitation experiments withellipsoidal particles, and observed behaviours that weredrastically different from those of spheres [3]. Below weshortly report the main experimental observations and weoffer a qualitative interpretation of the observed particlesdynamics.

2 Experimental hardwareThe particles were fabricated starting from

polystyrene spheres, 10 m in diameter. Prolate ellipsoids(average aspect ratio k > 1) were generated by uniaxialstretching, and oblate ones (k < 1) by biaxial stretching ofthe spheres. We thus obtained ellipsoids with dimensionsbetween a few m and several 10 m. We studied thebehaviours of these particles in a classical levitationexperiment, with a vertical laser beam, inside a glass cellfilled with water.

The beam waist diameter was 2.6 m, with acorresponding diffraction length ≈ 14 m.

3 ObservationsWe collected many observations with particles of

different aspect ratios located near the beam waist, in bulkwater (true levitation configuration) and when the particlewas pushed up to contact with the top boundary of thesample cell (a water-glass interface; in some experiments, a

water-air or a water-oil interface). Near the beam-waist,particles which were not too far from a sphere got trappedon the laser beam axis in static configurations, with theirlong axes lying vertical. Conversely, we found thatparticles of high ellipticity, either rod-like (k >> 1) or disk-like (k << 1) never came to static configurations: they wereseen to undergo sustained oscillations, in the form ofcoupled translation and tilt motions; see Fig. 1.

Sustained oscillations were observed in allcircumstances, be the particle in bulk or in contact to aboundary surface. The data evidence a bifurcatingbehaviour, from static to oscillating, at k ≈ 3 for prolateparticles and k ≈ 0.3 for oblate ones. We observedoscillations that were periodic (Fig. 2), bi-periodic orirregular (Fig. 3), depending on particle parameters andposition along the beam axis.

Figure 1 Top view of an oscillating oblate ellipsoid in bulkwater. The white cross ’X’ – represents the position of the beamcenter. The arrows represent the direction of oscillation.

3 Ray-optics modelWe propose an interpretation of the observed

dynamical behaviours on the basis of a simple 2-dimensional (2d) model for the interaction of an ellipsoidwith light, using the ray-optics (RO) approximation. A firstversion of the model, for prolate ellipsoids and a simplyparallel beam (as in Fig. 4 below), was reported in [3].

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For a given position and orientation of the ellipsoidwith respect to the beam axis (see Fig. 4), we compute theradiation pressure forces (F) and torques () experiencedby the particle – due to numerous momentum transfersthrough refraction and reflection– using standard ray-tracing algorithms. These forces and torques are then fedinto the equations of motion (written in the limit of smallparticle tilt angle, << /2) which are further integrated toaccess the particle dynamics. In the model we suppose thatthe optical forces F and torque Γ are balanced bycorresponding Stokes drag terms in translation androtation:

IIˆ 0m z F z g (1a)ˆ 0x F x (1b)

ˆ 0 Γ y (1c)

In Eqs. (1), x , z and are the axial, transverse andangular coordinates of the particle and the dot stands fortime derivative. II , and are the correspondingtranslational and rotational drag coefficients which areavailable in the literature for ellipsoids. This simple model

reproduces fairly well the observed bifurcation betweenstatic states and limit cycles when k increases.

Figure 3 Same as in Figure 2 but for a non-periodic signalobtained with a prolate ellipsoid located at the beam waist planewith k1=5.1 and k2=4.5.

Figure 4 2d ray optics (RO) model in the xz plane. The ellipsoidlong axis makes an angle with respect to the z-axis. 60reflections inside the ellipsoid are shown here for illustration.The laser beam has a Gaussian intensity profile.

In order to explore the effect of beam diffraction, werecently generalized the simulation to the case of adivergent/convergent beam striking the ellipsoid. We stillfind oscillations except that the particle dynamics nowdepends on the particle axial position ( z ). State diagramssummarizing the particle dynamics in the (k, z ) plane forvarious beam parameters will be presented. As a salientresult we find that much fewer oscillating configurations

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LASER-LIGHT AND INTERACTIONS WITH PARTICLES AUGUST 25-29TH, 2014, MARSEILLE, FRANCE

occur upon increasing the beam divergence. Occasionally,a bifurcation between two limit cycles emerges out of thecalculations (Fig. 5). We also computed force and torquemaps for various particle configurations (x,, z) andseveral ellipsoid aspect ratios (Fig. 6). These maps help usidentify the underlying k-dependent mechanismsresponsible for the onset of oscillations. In particular, theamplitude of the optical torque appears as a key parameterdriving the bifurcation.

Figure 5 Example of computed particle dynamics in phase spacefor two different initial conditions and for a convergent beamhitting the particle (see schematic on the right). The ellipsoidbifurcates here between two limit cycles.

As an essential outcome, our simple RO modelshows that the non linear dependence of the radiationpressure forces versus the particle position and anglecoordinates is good enough to reproduce the observedoscillations.

4 AcknowledgementThis work was financially supported by ITN Comploids

and Agence Nationale de la Recherche throughAMOCOPS project. We thank P. Merzeau, L. Buisson, P.Barboteau, E. Texier and J.Y. Juanico for technical help.

5 References[1] Jonáš A., Zemánek P., Light at work: the use of optical forcesfor particle manipulation, sorting, and analysis, Electrophoresis,29(24):4813-51 (2008)

[2] Ashkin A., Acceleration and Trapping of Particles by RadiationPressure, Phys. Rev. Lett. 24(4):156-9 (1970)

[3] Mihiretie B., Snabre P., Loudet J.C., Pouligny B., Radiationpressure makes ellipsoidal particles tumble, Europhys. Lett. 100,48005-1-6 (2012); Mihiretie B., Snabre P., Loudet J.C., Pouligny B.,Optical levitation and long-working distance trapping: from

spheres up to large aspect ratio ellipsoids, J. Quant. Spectrosc.Radiat. Transfer 126:61-68 (2013)

Figure 6 Exemplary maps of total force (Fx) and torque (y) as afunction of off-centering (x/R) and tilt () of the ellipsoid (TEpolarization mode; R is the radius of the mother sphere fromwhich the ellipsoid is derived). The ellipsoid is located below thebeam waist plane as shown in Fig. 3 (schematic). Aspect ratiok=4.25, beam waist radius w0=1.3 m, far-field beam divergenceangle: 5.4°.

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