Optical tweezers provide a non-invasive technique for manipulating microscopic-sized particles.  Electromagnetic waves transport energy and momentum.  When an electromagnetic wave interacts with a small particle, it can exchange energy and momentum with the particle.  The force exerted on the particle is equal to the momentum transferred per unit time.

The force exerted by optical tweezers is on the order of pN = 10^-12 N.  It is too weak to manipulate macroscopic-sized objects, but it is large enough to manipulate individual particles on a cellular level.  The force is distributed over most of the area of the particle, so fragile and delicate objects can be manipulated without causing damage.  Near-infrared laser beams can manipulate living cells without damaging them.

In early 1970s, Ashkin and coworkers demonstrated that micron-sized particles could be trapped, moved and levitated against gravity using two weakly focused laser beams.  A few years later, Ashkin demonstrated stable, three-dimensional trapping of dielectric spheres with a single, strongly focused the laser beam.  Such a trap is called a single-beam gradient force laser trap or "optical tweezers".  It is able to capture and manipulate particles, varying in size from several nanometers up to few tens of micrometers.  A typical setup is shown in the figure below.

A year later, Ashkin and Dziedzic used an argon-ion laser with a wavelength of 514 nm to demonstrate that optical tweezers are capable of trapping and manipulating living organisms.  However, light of this wavelength is strongly absorbed by bacteria and the bacteria are damaged at high power levels.  But biological cells and organisms are almost transparent to infrared light and a Nd-YAG laser with a wavelength of 1064 nm could be used to manipulate a variety of biological samples without damage.

Optical tweezers consists of a single laser beam, which is focused by a high-numerical-aperture objective lens to a diffraction-limited spot.

Assume that this laser beam is incident on a dielectric sphere with index of refraction slightly higher than the surrounding medium, floating in the surrounding medium.  If the radius of the sphere is much larger than the wavelength of the laser light (r ³ 10l), then we can use geometrical ray-optic to model the optical trap.  We assume that the wavelength of the beam is essentially zero, so that the trap focus becomes dimensionless and diffraction and interference effects become negligible.  In the geometrical approximation the incident laser beam can be represented by a set of parallel rays propagating in straight lines in a homogeneous medium, which are then focused ray-by-ray to a focal point.  Each ray is assumed to be a plane wave with photons having momentum p, where p is directed towards the focal point.  In non-absorbing media each ray can be reflected or refracted when it encounters the boundary between the dielectric sphere and the surrounding medium.  The Fresnel reflection and transmission coefficients depend on the polarization of the ray and the angle of incidence.

Reflection and refraction change the momentum of the incident photons and therefore transfer momentum to the sphere.  The total momentum transfer per unit time due to all reflected photons in the beam always has a component in the direction of propagation of the beam.  It is responsible for the radiation pressure.  A force in this direction is called a scattering force.  The total momentum transfer due to all refracted photons in the beam always points toward the focus of the laser beam.  A force in this direction is called a gradient force.  The gradient force points radially and axially towards the region of highest electric energy density (the focus).  Three-dimensional trapping results when the axial component of the gradient force is stronger than the axial component of the scattering force. Bringing the laser beam to a tight focus produces a strong axial gradient force.

Assume that a laser beam is incident from above the z = 0 plane in a material with index of refraction n₁ and comes to a focus at the origin.  If the laser beam has cylindrical symmetry, let the z-axis be the symmetry axis.  Two angles q₀ and f₀ then suffice to specify the path of a photon in the beam.

Assume a transparent sphere with radius R and index of refraction n₂ has its center located near the origin in the xz-plane.  Assume a photon is incident on the transparent sphere.  Assume that the radius of the sphere R is much larger than the wavelength of the photon l.  Assume that the photon is refracted and that the direction of the emerging photon is different from the direction of the incident photon.  Let p_f be the momentum the emerging photon and p_i the momentum of the incident photon.  The momentum transferred to sphere is p_i-p_f.  Let us calculate p_f given p_i.

If the center of the sphere lies on the z-axis, then all rays are meridional rays.  Let us trace a single meridional ray.  This is easiest done by choosing a coordinate system in which the center of the sphere lies at the origin, as shown in the figure below.

If the incident ray makes an angle q = q₀ + q_i with the z-axis then the exiting ray will make an angle q' = q₀ - q_i + 2q_r with the z-axis, where n₁sinq_i = n₂sinq_r.  If n₂ > n₁ then |q_i| > |q_r|.

If q_i > 0 the incident ray makes an angle q > q₀ with the z-axis.  Then q' < q.  The magnitude of the momentum of a photon of the ray is p = n₁hn/c = n₁h/l.  The z-component of the momentum of an incident photon is p_iz = -p cosq, and the z-component of the momentum of an exiting photon is p_fz = -p cosq’.  The z-component of the moment transfer to the sphere is Dp_z = p(cosq’ - cosq).  If the beam has cylindrical symmetry about the z-axis and is focused, then for q_i > 0 the focus lies on the z-axis above the origin and the force on the sphere exerted by the light because of refraction is upward, towards the focus.  |Dp_z| becomes larger when the angle q the ray makes with the z-axis becomes larger 

If q_i < 0 the incident ray makes an angle q < q₀ with the z-axis.  Then q' > q.  The z-component of the moment transfer to the sphere now is Dp_z = p(cosq’ - cosq) < 0.  If the beam has cylindrical symmetry about the z-axis and is focused, then for q_i < 0 the focus lies on the z-axis below the origin and the force on the sphere exerted by the light because of refraction is downward, towards the focus.  |Dp_z| becomes larger when the angle q the ray makes with the z-axis becomes larger.

The force on the sphere exerted by the light because of reflection is always downward, independent of the location of the focus.  To trap the particle we need the force exerted by refraction to have a larger magnitude than the force exerted by reflection when those two forces point in opposite directions.   |Dp_z| due to reflection becomes smaller when the angle q the ray makes with the z-axis becomes larger.

The force on the sphere exerted by the light because of reflection depends on the reflectance.  The reflectance is a function of the angle of incidence.  If n₂ is not very different from n₁ and the ratio n₂/n₁ is close to 1 than the reflectance is very close to zero over a wide angular range and trapping is possible with a beam that is tightly focused, i.e. with a beam that has a large q_max.  The two figures below show the reflectance as a function of the angle of incidence for two different values of n₁/n₂.

If the center of the sphere does not lie on the z-axis, then some rays are skew rays.  We can develop an algorithm that can be executed by a computer to find the force exerted by a laser beam on a dielectric sphere because of refraction.  We initially switch back to the coordinate system in which the laser comes to a focus at the origin. 

Assume that for any photon in the laser beam 0 £ q₀ £ q_max and 0 £ f₀ £ 2p.  The path of an incident photon can be described by the equations z = (1/tanq₀)r, f = f₀.

The magnitude of the momentum of an incident photon is p = n₁hn/c = n₁h/l.  The components of its momentum vector are p_z = -pcosq₀, p_x = -psinq₀cosf₀, and p_y = -psinq₀sinf₀.

Assume the center of a sphere of radius R is located at r₀ = (x₀,y₀,z₀).  Let us calculate the change in the photon’s momentum if its path crosses this sphere.  We assume that the sphere can absorb this change in momentum with negligible change in energy, i.e. that it is "infinitely heavy".

Procedure:

The algorithm consists of a series of rotations and a translation.  If we let the z-axis of the coordinate system denote the optic axis, then a skew ray entering a sphere in one coordinate system can become a meridional ray in rotated and translated coordinate system. We can calculate the momentum transferred to the sphere because of refraction of the meridional ray as outlined above.  Then we reverse the order of the rotations of the coordinate system to find the momentum transfer in the original coordinate system.

Optical tweezers use a single laser beam.  They are easy to construct and use.  Once a particle is trapped, it can be moved around in the surrounding medium without mechanical contact and without the risk of contaminating a sample which is in a closed, sterilized chamber.  Optical trapping is a powerful tool.  Its usefulness can be increased when optical tweezers are combined with optical scissors or scalpels for cell microsurgery.  These devices consist of UV or visible laser microbeams, which are used for precision cutting and drilling of samples.

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