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Theoretical determination of the radiation force for a spherical particle illuminated by a focused laser beam
Abstract
Trapping forces on dielectric spheres in single beam laser tweezers are computed. A focused beam description based on an exact solution of Maxwell's equations is compared to the 5th order Gaussian beam approximation due to Barton and Alexander. Forces on water droplets suspended in air and on polystyrene spheres suspended in water, exerted by beams focused to varying degree, are calculated. It is demonstrated that the 5th order approximation is accurate for almost paraxial beams (numerical aperture NA <∼0.25), as compared to the exact treatment. However, for strongly focused beams the 5th order approximation breaks down. Thus it is established that an accurate beam description is vital for modeling optical traps, since, in order to hold a particle effectively in a single beam trap, a strongly focused beam is required.
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Starting from a Debye-type integral representation valid for a laser
beam focused through a high numerical aperture objective, we derive an
explicit partial-wave (Mie) representation for the force exerted on a
dielectric sphere of arbitrary radius, position and refractive index. In
the semi-classical limit, the ray-optics result is shown to follow from
the Mie expansion, holding in the sense of a size average. The
equilibrium position and trap stiffness oscillate as functions of the
circumference-to-wavelength ratio, a signature of interference, not
predicted by previous theories. We also present comparisons with
experimental results.
We derive and apply formulas that employ the vector-wave addition theorem and rotation matrices for quan-titative calculations of both radial and axial optical forces exerted on particles trapped in arbitrarily shaped tweezer beams. For the tightly focused beams encountered in optical tweezers, we shall highlight the impor-tance of formulating the optical forces and beam symmetries in terms of the irradiance and total beam power. A major interest of the addition theorem treatment of optical forces is that it opens up the possibility of mod-eling a wide variety of beam shapes while automatically ensuring that the beams satisfy the Maxwell equa-tions. In some of the first numerical applications of our method, we shall illustrate that resonance effects play an important role in the axial trapping position of particles comparable in size with the wavelength of the trapping beam.
An investigation is made of the structure of the electromagnetic field near the focus of an aplanatic system which images a point source. First the case of a linearly polarized incident field is examined and expressions are derived for the electric and magnetic vectors in the image space. Some general consequences of the formulae are then discussed. In particular the symmetry properties of the field with respect to the focal plane are noted and the state of polarization of the image region is investigated. The distribution of the time-averaged electric and magnetic energy densities and of the energy flow (Poynting vector) in the focal plane is studied in detail, and the results are illustrated by diagrams and in a tabulated form based on data obtained by extensive calculations on an electronic computor. The case of an unpolarized field is also investigated. The solution is not restricted to systems of low aperture, and the computational results cover, in fact, selected values of the angular semi-aperture alpha on the image side, in the whole range 0
Micron-sized particles have been accelerated and trapped in stable optical potential wells using only the force of radiation pressure from a continuous laser. It is hypothesized that similar accelerations and trapping are possible with atoms and molecules using laser light tuned to specific optical transitions. The implications for isotope separation and other applications of physical interest are discussed.
Numerical computations in the framework of the generalized Lorenz–Mie theory require the evaluation of a new double set of coefficients g n , TM m and g n , TE m (n = 1, …, ∞; m = − n, … +n). A localized interpretation of these coefficients is designed to permit fast and accurate computations, even on microcomputers. When the scatter center is located on the axis of the beam, a previously published localized approximation for a simpler set of coefficients gn is recovered as a special case. The subscript n in coefficients gn and g n m is associated with ray localization and discretization of space in directions perpendicular to the beam axis, while superscript m in coefficients g n m is associated with azimuthal wave modes.
Fifth???order corrected expressions for the electromagnetic field components of a monochromatic fundamental Gaussian beam (i.e., a focused TEM 0 0 mode laser beam) propagating within a homogeneous dielectric media are derived and presented. Calculations of relative error indicate that the fifth???order Gaussian beam description provides a significantly improved solution to Maxwell???s equations in comparison with commonly used paraxial (zeroth???order) and first???order Gaussian beam descriptions.
Series expressions for the net radiation force and torque for a spherical particle illuminated by an arbitrarily defined monochromatic beam are derived utilizing the spherical‐particle/arbitrary‐beam interaction theory developed in an earlier paper. Calculations of net force and torque are presented for a 5‐μm‐diam water droplet in air optically levitated by a tightly focused (2 μm beam waist diameter) TEM 0 0 ‐mode argon‐ion (λ=0.5145 μm) laser beam for on and off propagation axis, and on and off structural resonance conditions. Several features of these theoretical results are related to corresponding experimental observations.
Organelle transport along microtubules is believed to be mediated by organelle-associated force-generating molecules. Two classes of microtubule-based organelle motors have been identified: kinesin and cytoplasmic dynein. To correlate the mechanochemical basis of force generation with the in vivo behaviour of organelles, it is important to quantify the force needed to propel an organelle along microtubules and to determine the force generated by a single motor molecule. Measurements of force generation are possible under selected conditions in vitro, but are much more difficult using intact or reactivated cells. Here we combine a useful model system for the study of organelle transport, the giant amoeba Reticulomyxa, with a novel technique for the non-invasive manipulation of and force application to subcellular components, which is based on a gradient-force optical trap, also referred to as 'optical tweezers'. We demonstrate the feasibility of using controlled manipulation of actively translocating organelles to measure direct force. We have determined the force driving a single organelle along microtubules, allowing us to estimate the force generated by a single motor to be 2.6 x 10(-7) dynes.
The development of the gradient force optical particle trap ('optical tweezers') has made it possible to manipulate biological materials using a single beam of laser light. Optical traps can produce forces in the microdyne range on intact cells without causing overt damage: such forces are sufficient to arrest actively swimming bacteria and can overcome torque generated by the flagellar motor of a bacterium tethered to a glass surface by a flagellar filament. By calibrating the trapping force against Stokes' drag and measuring the twist that is sustained by this force, we determined the torsional compliance of flagella in tethered Escherichia coli and a motile Streptococcus. Flagella behaved as linear torsion springs for roughly half a revolution, but became much more rigid when turned beyond this point in either direction.
The force generated by the radiation pressure of a low power laser beam induces an optical trap which may be used to manipulate sperm. We studied the effect of the optical trap on sperm motility. A Nd:YAG laser beam was coupled to a conventional microscope and focused into the viewing plane by the objective lens. Sperm were caught in the trap and manipulated by a joy stick controlled motorized stage. After different exposure periods, the velocity and patterns were analysed by a computerized image processor. There were minor changes in sperm velocity when exposed to the trap for 30 seconds or less. A gradual decrease in the mean linear velocity was observed after 45 seconds of exposure. This optical micromanipulator may also be useful for studying the force generated by a single spermatozoa and evaluating the influence of drugs on motility.
A family of closed-form expressions for the scalar field of strongly focused Gaussian beams in oblate spheroidal coordinates is given. The solutions satisfy the wave equation and are free from singularities. The lowest-order solution in the far field closely matches the energy density produced by a sine-condition, high-aperture lens illuminated by a paraxial Gaussian beam. At the large waist limit the solution reduces to the paraxial Gaussian beam form. The solution is equivalent to the spherical wave of a combined complex point source and sink but has the advantage of being more directly interpretatable.
Computation of Forces exerted on a Microparticle by a Laser Beam
- R R Dorizzi
Dorizzi, R. R., " Computation of Forces exerted on a Microparticle by a Laser Beam, " PhD thesis,
University of Hertfordshire (2004).
Modelling laser entrapment forces on microspheres
- N T Jones