IntroductionOptical Traps

	Schematic animation of the trapping of a polysterene bead with a focused single laserbeam trap (optical tweezer). (Animation by Steve Pawlizak, 2009.)

In case of biological samples, light in the near infrared (700-1100 nm) is used in order to prevent radiation damage, which would occur with shorter wavelengths. The different processes responsible for this "opticution" are a subject of controversy.

The list of experiments for which optical tweezers have been used includes the trapping of cells and bacteria [3,4], the measuring of the forces exerted by molecular motors such as myosin or kinesin [5,6] and the study of the machanical properties of single DNA strands fully expanded by the means of beads attached to their ends [7]. A good overview of biological applications is given in [8].
 

Theoretical Description

In general, scattering, i.e. the interaction of light with an object, can be divided into two components [9]. The first is the reflection and refraction at the surface of the particle, the second is the diffraction from the rearrangement of the wavefront after it interacts with the particle. While the radiation pattern due to reflection and refraction emanates from the particle in all directions and depends on the refractive index of the particle, the diffraction pattern is primarily in the forward direction and depends only on the particle geometry.

Two different regimes of theoretical approach can be distinguished. They are determined by the ratio of the incident light's wavelength λ to the diameter D of the irradiated particle. In the ray optics regime regime, the particle is very large compared to the wavelength (D >> λ), whereas in the Rayleigh regime the opposite is true (D << λ). The calculation of optical forces for arbitrary particle sizes D ≈ λ is nontrivial. For a full arbitrary theory, the solution of Maxwell's equations with the appropriate boundary condition is required [9]. The Lorenz-Mie Theory [10] was the first step into that direction and describes scattering of a plane wave by a spherical particle for arbitrary particle size, refractive index and wavelength. However, the Lorenz-Mie Theory cannot describe a Gaussian beam, such as that produced by a TEM₀₀ laser, which is a particular point of interest to accurately describe laser-induced forces. The calculation of optical forces of Gaussian beams and arbitrarily shaped beams can be achieved by the Generalized Lorenz-Mie Theory (GLMT) [11, 12]. 
 
 

Rayleigh Regime (D << λ)

In the Rayleigh regime, the particle is very small compared to the wavelength (D << λ). The distinction between the components of reflection, refraction and diffraction can be ignored. Since the perturbation of the incident wavefront is minimal, the particle can be viewed as an induced dipole behaving according to simple electromagnetic laws.

Scattering Force:

One of the two arising forces is the scattering force due to the radiation pressure on the particle. Incident radiation can be absorbed and isotropically reemitted by atoms or molecules. With this, two impulses are received by the molecule, one along the beam propagation of the incident light and one opposite to the direction of the emitted photon. Since the photon emission has no preferred direction, a net force results in the direction of incident photon flux. This force is directed along the propagation of light and is given by
 

(1)

where n_m is the refractive index of the surrounding medium, <S> is the time averaged Poynting vector, c is the speed of light, and σ is the particle's cross section, which in case of a spherical particle is given by
 

(2)

with the particle radius r, the refractive index n of the particle and the wavevector k of the used light.

Gradient Force:

The second force arising is the gradient force. This force is due to the Lorenz force acting on the dipole, induced by the electromagnetic field. In the field of the laser, the gradient force on an induced dipole
 

(3)

of polarizability α is given by [13]
 

(4)

where  is the electric field vector of the laser light. With the identity
 

(5)

and the results of Maxwell's equations
 

(6)

eq. (4) becomes
 

(7)

The gradient force which the particle experiences is the time-averaged version. The relation

yields
 

(8)

The light intensity I associated with the field is [13]
 

(9)

where n_m is the refractive index of the exposed material, ε₀ is the permittivity of free space, and c is the speed of light. This leads to the force relation
 

(10)

The gradient force's direction is toward the area of highest light intensity, i.e. toward the beam axis in case of a Gaussian beam profile, and toward the focus of the laser if the beam is focused.
 

Fig. 1: The forces arising in the Rayleigh regime for a slightly diverging laser beam.

A particle displaced from the beam axis experiences the gradient force as a restoring force toward the beam axis. Due to the low curvature of the beam, a component of the gradient force parallel to the direction of light propagation can be neglected. The scattering force due to radiation pressure pushes the particle in the direction of light propagation. For the trapping of particles, either two counterpropagating beams are required in order for the scattering forces to cancel out (geometry of the "Optical Stretcher"), or a single laser beam has to be focused very tightly.
 

Fig. 2: The forces arising in the Rayleigh regime for such a tightly focused laser beam ("Optical Tweezers").

In a highly focused beam, the gradient force possesses a component against the Poynting vector in addition to its component perpendicular to it. This component prevents the particle from being pushed in the direction of light propagation by the scattering force. The net force acts as a restoring force toward the focus of the beam with respect to all three dimensions.
 
 

Ray Optics Regime (D >> λ)

In the ray optics regime, the size of the object is much larger than the wavelength of the light, and a single beam can be tracked throughout the particle. If the ratio of the refractive index of the particle to that of the surrounding medium is not close to one but sufficiently large, diffraction effects can be neglected, which is thoroughly explained in [9] and assumed for further discussion. This situation is for example given when whole cells, which are microns or tens of microns in size, are trapped using infrared light while suspended in solution. The incident laser beam can be decomposed into individual rays with appropriate intensity, momentum, and direction. These rays propagate in a straight line in uniform, nondispersive media and can be described by geometrical optics.

Consider a Gaussian laser beam hitting a spherical particle of refractive index n₁, which is surrounded by a medium of refractive index n₂ (Fig. 3a). 
 

Fig. 3: The momentum (red arrows) of (a) one ray and (b) two rays with different intensities propagating through a sphere. The blue arrow indicates the restoring net force.

Each incoming ray carries a certain amount of momentum p proportional to its energy E and to the refractive index n_i of the medium it travels in:
 

(11)

After a light ray traveled through the particle, its momentum has changed in direction and magnitude. The momentum difference is picked up by the particle. The force due to the directional change of a ray's momentum has components in the forward direction as well as to the side. However, there are many rays incident on the particle. The net force has only a forward component due to the rotational symmetry of the problem. This symmetry is broken if the particle is not centered exactly on the optical axis of the Gaussian beam. In this case the particle feels a restoring force (Fig.3b).

The net force's component perpendicular to the beam propagation is called the gradient force, its component along the direction of beam propagation the scattering force, in reference to Rayleigh scattering (Fig. 1). In the ray optics regime, however, these two force components stem from just one single physical effect as discussed.

At closer inspection, it becomes obvious that forces are actually applied at discontinuities in refractive indices, i.e. at the surface. The net force is due to the combination of all surface forces. For a rigid object, such as a glass bead, the net force is the only force that matters. For a soft object, the forces on the surface become important and lead to a deformation of the object [14]. This is the basis for a novel optical tool - the "Optical Stretcher".
 

References:
 

[1]		A. Ashkin: Acceleration and trapping of particles by radiation pressure, Phys. Rev. Lett. 24:156-159 (1970)
[2]		A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu: Observation of a single-beam gradient force optical trap for dielectric particles, Opt. Lett. 11:288-290 (1986)
[3]		A. Ashkin, J. M. Dziedzic, T. Yamane: Optical Trapping and Manipulation of Single Cells Using Infrared Laser Beams, Nature 330:769-771 (1987)
[4]		A. Ashkin, J. M. Dziedzic: Optical Trapping and Manipulation of Viruses and Bacteria, Science 235:1517-1520 (1987)
[5]		R. M. Simmens, J. T. Finer, H. M. Warrick, B. Kralik, S. Chu, J. A. Spudich: Force on Single Actin Filaments in a Motility Assay Measured with an Optical Trap. In: G. H. Pollack, H. Sugi (Eds.): The Mechanism of Myofilament Sliding in Muscle Contraction, Plenum, New York (1993), pp. 331-336
[6]		S. C. Kuo, M. P. Sheetz: Force of single kinesin molecules measured with optical tweezers, Science 260:232-234 (1993)
[7]		S. Chu: Laser Manipulation of Atoms and Particles, Science 253:861-866 (1991)
[8]		K. Svoboda, S. M. Block: Biological applications of optical forces, Annu. Rev. Biophys. Biomol. Struct. 23:147-285 (1994)
[9]		H.C. v.d. Hulst (Ed.): Light Scattering by Small Particles, Dover, New York (1981)
[10]		G. Mie: Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen, Ann. Phys. 25:377-452 (1908)
[11]		G. Gouesbet: Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres, J. Opt. Soc. Am. 16:1641-1650 (1999)
[12]		G. Gouesbet, B. Maheu, G. Grehan: Light scattering from a sphere arbitrarily located in a gaussian beam, using a Bromwich formulation, J. Opt. Soc. Am. 5:1427-1443 (1988)
[13]		Y. Harada, T. Asakura: Radiation forces on a dielectric sphere in the Rayleigh scattering regime, Opt. Commun. 124:529-541 (1996)
[14]		J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, J. Käs: Optical deformability of soft biological dielectrics, Phys. Rev. Lett. 84:5451–5454 (2000)