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Non Conservative Optical Forces for Silicon Nanowires in Optical Traps NON CONSERVATIVE OPTICAL FORCES FOR SILICON NANOWIRES IN OPTICAL TRAPS A. MAGAZZU1,2,*, A. IRRERA1, P. ARTONI3, S. H. SIMPSON4, S. HANNA4, P. H. JONES5, F. PRIOLO3, P. G. GUCCIARDI1, and O. M. MARAGÓ1,* 1 CNR-IPCF, Istituto per i Processi Chimico-Fisici, Messina, Italy 2 Dottorato in Fisica, Università di Messina , Messina, Italy 3 Matis CNR-IMM and Dipartimento di Fisica, Università di Catania, Catania, Italy 4 H. H. Wills Physics Laboratory, University of Bristol, Bristol, UK 5 Department of Physics and Astronomy, University College London, London, UK *Corresponding authors: [email protected], [email protected] Abstract For a non-spherical particle the non-conservative forces We measure non-conservative forces in optical can be considered composed by two contributions. A first trapping of ultra-thin Silicon nanowires by photonic force contribution depends on the standard non-homogeneous and torque microscopy. We reveal how the extreme non- radiation pressure due to the gaussian geometry of the spherical shape generates a transverse component of the laser beam that may yield a rotational force in ρ-z plane radiation pressure that results in a thermally activated even for a spherical trapped particle [4,5]. The second non-conservative rotation of the nanowire about the trap contribution, manifest for nonspherical objects, depends axis. We explore the behavior with trapping power and on alignment along the axial direction, that generates a scaling with nanowire length. This has implications for transverse force (optical lift effect) [6] that causes rotations optical force calibration and optomechanics with levitated and precessions in the ρ-z plane. non-spherical particles. 1 Introduction Optical tweezers (OT) are powerful tools to understand, detect and measure forces in soft-matter systems at the micro and nanoscale [1]. They are commonly used in areas as diverse as biophysics, colloidal physics, and hydrodynamics of small systems. One of the crucial enabling step of OT is the accurate calibration of the applied optical forces. An implicit assumption of all calibration methods is that, for small displacements of a particle from the center of a Gaussian beam optical trap, the restoring force is proportional to the displacement, i.e., the optical trap is assumed to act as a Hookeian spring with a fixed stiffness. This condition implies that the optical force-field produced by optical forces must be conservative, excluding the possibility of a rotational (non-conservative) component [2]. This is not strictly correct since scattering forces are intrinsically non- conservative and this has some measurable consequences. Figure 1 depiction of a SiNW in an optical trap and forces acting on it. 2 Optical forces For a generic particle in optical tweezers the optical forces are essentially the sum of two types: the gradient 3 Non-conservative effects (conservative) force is responsible for trapping [2], while For an optically trapped spherical particle, non the non-conservative part generally has a destabilizing conservative effects are only experienced when the laser effect [3,4,5]: power, and hence optical potential depth, is sufficiently low to permit the (thermally fluctuating) sphere to F = -U + FNC explore a large trapping volume, and to feel the inhomogeneity of the scattering force which is greater ME-16.1 LASER-LIGHT AND INTERACTIONS WITH PARTICLES AUGUST 25-29TH, 2014, MARSEILLE, FRANCE nearer the edges of the optical potential well. Under these We conclude that non-conservative effects on non- conditions the thermal motion displays a toroidal bias spherical particles are linked to the occurrence of a coupling the directions transverse (ρ) and parallel (z) to transverse component of the radiation pressure. Although the beam propagation axis. By increasing the laser power we measured non-conservative effects in the order of few the non-conservative effects tend to vanish, being of percent of the trap dynamics, we were able to detect them second-order with respect to the gradient (conservative) at any available laser power, implying that this needs to forces [7]. However for non-spherical particles, such be take into account when dealing with calibration of effects have been predicted to be first-order, hence non- optical trapping forces based on effective potentials vanishing with the laser power and affecting optical force [11,12]. Moreover non-conservative forces can have a calibration. more dramatic effect when dealing with optomechanics of In the case of nanowires increasing the length of the levitated non-spherical particles [1]. SiNWs the transverse stiffness tends to L-3. Instead the rotational term, initially increases with L for parts of SiNWs outside the focus point and when L is great enough, to cause the orientation and constrainment of the nanowire in the transverse direction, it tends to L-1 [8]. The rotational term decreases slower and is more dominant than the translational term. For this reason we can infer that non conservative rotation in xy plane is a roto-translation of the nanowire about the optical axis, rather than a precession of the wire about the centre-of- mass point. Here we report the first observation of non- conservative effects on non-spherical particles, specifically Figure 2 Non-conservative component of the force-field as on ultra-thin silicon nanowires (SiNWs) with a diameter function of the SiNWs length. of 10 nm and several lengths (L = 1.00±0.05, 2.08±0.04, 3.1±0.1,4.0±0.2 and 5.0±0.5 μm) [9,10]. These SiNWs are 4 References held in an optical tweezers where optical forces align the [1] O. M. Maragò, P. J. Jones P. G. Gucciardi, G. Volpe et al, axis of the NW close to the propagation direction of the Nature Nanotechnology, 8, 807-819 (2013) trapping laser beam as shown in Figure 1. [2] Ashkin, A., Dziedzic, J., Bjorkholm, J., and Chu, S. Opt. Lett. We show how the non-conservative effects depend on 11, 288 (1986). shape and size of the trapped particle and on laser power. We detect two different kinds of thermal motion in the [3] Volpe Giorgio, V. G. and Petrov, D. Physical Review E 76, 061118 trap, a toroidal motion coupling the ρ-z directions and a (2007). roto-translation coupling the x-y directions, Figure 1. We [4] David G. Grier et al. Physical Review Letters 101, 128301 (2008) characterize the two dimensional motion of SiNWs, by the [5] G. Pesce, G. Volpe, Europhysics Letters, 86 38002, (2009) Differential Cross Correlations (DCCFs) [3] of particle tracking signals, that highlight roto-translation and [6] Simpson, S. H., Hanna, S., Peterson, T. J., and Swartzlander, G. A. toroidal bias. We explore the scaling behavior of these non Optics Letters 37(19), 4038–4040 (2012). conservative effects with trapping power and with [7] Simpson, S. H. and Hanna, S. J. Opt. Soc. Am. A 27(9), 2061–2071 nanowire length Sep (2010). We also calculated the non conservative component of [8] Simpson, S. and Hanna, S. Nanotechnology 23(20),205502 the force-field, shown in Figure. 2, that takes account of (2012). the amount of the non conservative effects and is defined as the ratio of the rotational and the average of the [9] Irrera, A. and et al. Nano Lett. 11, 4879–4884 (2011). relaxation frequencies in the x-y plane:, ij = ij/. [10]Irrera, A., Artoni, P., Iacona, F., Pecora, E., Franzo, G., Galli, Typically the non-conservative contribution to the M., Fazio, B., Boninelli, S., and Priolo, F. Nanotechnology 23(7), trapping force is of the order of a few percent of the total 075204 (2012). optical force acting on the particle. [11]Nieminen, T., Rubinsztein-Dunlop, H., and Heckenberg, N. We find that the non-conservative contribution scales Journal of Quantitative Spectroscopy and Radiative Transfer 70(4), 627– as 1/L represented in figure 2 by the black line assuming 637 (2001). large L, L>>zR where zR is the Rayleigh range of the strongly focused trapping laser beam which measures the [12]Bishop, A. I., Nieminen, T. A., Heckenberg, N. R., and Rubinsztein- length of the focal region parallel to the beam axis. Dunlop, H. Physical Review A 68(3), 033802 (2003). ME-16.2.
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