Optical trapping forces depending on size of dielectric polystyrene microspheres

Abstract

A single-beam gradient force trap or an optical laser tweezer has been a powerful tool to accurately measure pico-newton to femto-newton forces in a variety of fields, such as biology, colloid/polymer physics, and rheology. When an appropriate refractive index mismatch between an object and a medium is provided, the laser beam can hold the object of which its size ranges from a few nanometers to tens of micrometer scale. The optical trapping mechanism depends on the relative length scale between an object and the wavelength of a light. In the Rayleigh regime where the diameter of the object is smaller than the wavelength of the light ( ), the trap stiffness increases with the particle size, as the number of rays incident on the particle increases. On the contrary, all rays hit an object in the ray optics regime ( ) such that Ashkin expected the trapping force to be independent of its size without quantitative calculations. Similarly, Wright et al. predicted that the size dependence on the axial and radial trapping forces are negligible in the ray optics regime, based on calculating the dimensionless efficiency of the optical trapping force for a single ray. Using the generalized LorenzMie theory, Lock also suggested that the trapping force in the axial direction does not change with the particle size. However, Simmons et al. reported the experimental results in which the trap stiffness for dielectric particles decreases as the size increases when the particle size is larger than the wavelength of a light. More recently, Mazolli et al. used an explicit partial-wave representation to extend the force expression from the Rayleigh regime (i.e., electromagnetic force model) to the GOA. In the ray optics regime, they reported that the trap stiffness is inversely proportional to the particle radius. Tlusty et al. also calculated the trapping force of highly localized fields on dielectric particles in which the trapping force in the ray optics regime decreases as their size increases. In this communication, we aim to address such contradiction, regarding the dependence of the optical trapping force on the particle size in the ray optics regime. Based on the GOA, we analytically calculate the optical trapping force with varying the particle size. Then, we compare the analytical calculations to the drag experiments in which a particle trapped by the laser tweezers is subjected to the Stokes drag force. We also provide a schematic explanation, based on drawing the ray propagation and finding the magnitude of the momentum change. Before we report our results, experimental materials and method are briefly described. We use polystyrene (PS, Interfacial Dynamics Corporation) latex particles with different radii, a=1.4, 1.7, and 3.7 μm. The refractive index of these particles is np=1.57. In order to obtain the trap stiffness (κt), which is the ratio of the Stokes drag force (Fs=6πaηu) to the lateral displacement (∆y) from the center of the particle, it is held in the aqueous phase (refractive index, nm=1.326) by the laser tweezers, and subjected to drag forces by translating the microscope stage at several constant velocities between u=3 and 7 μm/s. The viscosity of the aqueous solution is η=1 cP. We use particle tracking to measure the particle displacement during applying the Stokes flow around the particle. A movie captured on a CCD camera (Hitachi KP-M1) is saved at a frame rate of 29.97 frames/s. The digitized image sequences are imported to image processing software (ImageJ) to find the particle displacement (∆y) from the equilibrium position in the optical trap. The laser tweezer apparatus is constructed around the inverted microscope. The laser is generated by a 4-Watt CW Nd:YAG laser (λ0=1,064 nm, Coherent Compass 1064-400M). A water immersion objective (63×, NA 1.2, Zeiss C-Apochromat) is used to provide a high convergence angle (φmax), which is a maximum half angle of the conelike laser beam, given by φmax=sin(NA/nm)≈64.5. We refer the reader to Pantina and Furst for a more detailed description of the laser tweezer setup. In order to quantitatively calculate the optical trapping force in the ray optics regime, we consider the first reflection and the successive refractions of a single incident ray with power (P) hitting a dielectric spherical particle, the dimensionless efficiency (Q) of the trapping force is given by,