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Chapter 8
Optical Vortex Trapping and the Dynamics of Particle Rotation Timo A. Nieminen, Simon Parkin, Theodor Asavei, Vincent L.Y. Loke, Norman R. Heckenberg, and Halina Rubinsztein-Dunlop University of Queensland, Australia http://www.download-it.org/learning-resources.php?promoCode=&partnerID=&content=story&storyID=1899
8.1 INTRODUCTION Since the pioneering work by He and colleagues [34,35], the transport of optical angular momentum by optical vortex beams has been applied to the rotation of microscopic particles in optical traps. Even earlier, optical vortex traps had been proposed as a method to reduce the scattering forces that oppose optical trapping; in the ray optics picture, only high-angle rays contribute to the gradient force, and the use of a hollow beam eliminates the low-angle rays that would still contribute to the scattering force [1]. There are many interesting phenomena in trapping and micromanipulation of microscopic particles using optical vortex beams; some depend on the transport of orbital angular momentum by the beam, and others do not. It is not easy to predict the behavior of conventional optical traps other than in a very general way. To go beyond that to accurate quantitative results (and sometimes the revelation of behavior that is qualitatively surprising as well) requires computational modeling of the interaction between trapping beam and trapped particle. While methods such as geometric optics and Rayleigh scattering are only accurate in large and small particle regimes outside the usual range of particles trapped and STRUCTURED LIGHT AND ITS APPLICATIONS Copyright © 2008 by Elsevier Inc. All rights of reproduction in any form reserved.
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Optical Vortex Trapping and the Dynamics of Particle Rotation
manipulated [1,33], it is possible to use electromagnetic theory to model optical trapping in the gap between the domains of these methods [60,64–66,71]. In practice, it is possible to obtain agreement with experiment to better than one percent [43]. After an initial review of theoretical basics, and the properties of nonparaxial optical vortices, we will computationally investigate a variety of different phenomena in optical vortex trapping and micromanipulation, review related experimental results, and discuss possible further experimental work.
8.2 COMPUTATIONAL ELECTROMAGNETIC MODELING OF OPTICAL TRAPPING The calculation of optical forces and torques is essentially an electromagnetic scattering problem—the incident field carries energy, momentum, and angular momentum toward the particle in the trap, and the superposition of the scattered and incident fields carries these away. The difference between the inward and outward fluxes gives the absorbed power, and optical force and torque. As noted above, particles typically http://www.download-it.org/learning-resources.php?promoCode=&partnerID=&content=story&storyID=1899 trapped and manipulated using optical tweezers are inconveniently both too small for short wavelength approximations such as geometric optics and too large for long wavelength approximations such as Rayleigh scattering. There are many different methods available for the computational modeling of electromagnetic problems in this intermediate size range. Perhaps the most widely used are the finite-difference time-domain method (FDTD) and the finite-element method (FEM). Due to the popularity and versatility of such methods, it is remarkable that so little use has been made of them for modeling optical trapping. On a closer examination of the problem, the reasons for this become clear. First, repeated calculations are needed for modeling trapping—perhaps a few dozen calculations to find the equilibrium position of a particle within a trap, and the spring constant, through to thousands when, for example, calculating a “map” of the force exerted on the particle as a function of axial and radial displacement in the trap. Second, the typical optical tweezers arrangement provides us with a relatively simple electromagnetic problem—there is one particle, usually spherical and on the order of a few wavelengths in size, scattering a monochromatic beam, far enough away from surfaces so that they can be safely ignored. This is not far removed from the Lorenz–Mie problem, the scattering of a plane wave by a single sphere, for which an analytical solution exists. The basic formulation of the Lorenz–Mie solution is very simply extended to arbitrary monochromatic illumination, or even to nonspherical particles (while the theoretical extension is simple, the practical extension is another matter; some of the issues
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Computational Electromagnetic Modeling of Optical Trapping
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involved are considered later). Fundamentally, the Lorenz–Mie solution makes use of (inc) a discrete basis set of functions ψn , where n is mode index labeling the functions, each of which is a divergence-free solution of the Helmholtz equation, to represent the incident field, Uinc =
∞ n
an ψn(inc) ,
(1)
and ψk(scat) to represent the scattered wave, so that the scattered field can be written as Uscat =
∞ k
(scat)
p k ψk
.
(2)
The expansion coefficients an and pk together specify the total field external to the particle. When the electromagnetic response of the scatterer is linear, the relationship between the incident and scattered fields must be linear, and can be written as the matrix equation http://www.download-it.org/learning-resources.php?promoCode=&partnerID=&content=story&storyID=1899
pk =
∞
Tkn an
(3)
n
or P = TA.
(4)
The Tkn , which are the elements of the transition matrix, or system transfer matrix, often simply called the T-matrix, are a complete description of the scattering properties of the particle at the wavelength of interest. This is the foundation of the classic Lorenz–Mie solution [51,54], the extension of the Lorenz–Mie solution to arbitrary illumination, usually called generalized Lorenz–Mie theory (GLMT) [28], the extension of the Lorenz–Mie solution to nonspherical but still separable geometries such as spheroids, also usually called GLMT [31,32], and general problem considered for an arbitrary particle and arbitrary illumination, usually called the T -matrix method [55, 69,89]. When the scatterer is finite and compact, the most useful set of basis functions is vector spherical wave functions (VSWFs) [55,69,70,89]. In particular, the convergence of the VSWFs is well-behaved and known [9], and this allows the sums given above to be truncated at some finite nmax without significant loss of accuracy. At this point, we can outline the basic procedure for calculating the optical force and torque acting on a particle in a trap: 1. Calculate the T -matrix
Chapter extract
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