Analysis of light wave scattered by an optically trapped particle

Abstract

The laser trapping technique(1) allows one to study a single particle in a noncontact manner and to even manipulate biological particles without giving damage on them if a light of specific optical wavelength is used. In the technique a focused laser beam is necessary to give a trapping force on the particle. The force is calculated by ray tracing for particles from several microns to tens of microns. However, there exists a problem whether it is appropriate to do ray tracing in the range where geometrical optics is no more applicable. On the other hand, light scattered by particle smaller than a micron can be described by Mie theory which is fully based on electromagnetic equations. If the scattering pattern can be obtained, the force on the particle can be determined without ray tracing. So far, scattering patterns by Mie theory has been solved for plane wave illumination. In this paper, we describe a method to calculate the scattering patterns by a convergent beam illumination, which is the case for laser trapping, and applied to study various particles dissolved in liquid using the scattered patterns.Scattering pattern from a spherical particle with plane wave illumination is described by Mie theory, and we can calculate the scattered wave for any specific direction. A focused beam having a wavefront, which could either be given or measured experimentally, can be expanded to a series of plane waves by Fourier transformation. The scattered pattern of each incident plane wave is calculated in a two dimensional array for equal interval of angle or its sine function. The total scattered pattern is vector superposition of these single scattering patterns. Since we observe the back-scattered light in the experiments, the scattered pattern is calculated in the array for equal sine values. The scattering pattern with a convergent beam was calculated by supposing a Gaussian distribution of the wavefront The convergent beam is considered as a combination of various plane waves having different phase and amplitude and the total scattered wave is calculated as the superposition of those plane waves by Mie theory.A Nd:YAG laser is collimated and coupled to the optical microscope through a half transparent mirror. The objective lens with numerical aperture of 0.9 was used to focus the laser beam and therefore traps the particle near its focus. The back-scattered light from the particle passed through the objective and another camera lens was captured by a CCD camera which can be fed into a computer. We found the scattered patterns are dependent on the size and shape of the particle trapped and is quite different to the pattern that Mie theory with plane wave illumination predicts. We found that the pattern calculated with the Gaussian beam assumption for convergent illumination is much closer to the pattern observed in the experiments. We think the scattering pattern can give information on the particle structure as well as trapping force on the particle.