Biophotonics_Concepts_to_Applications

this focused beam, which is called thebeam waist. Within this region, the electric
field is strongest in the center of the beam. Thus if a particle is displaced from the
center of the beam by a distance x, the gradient force will push the particle back to
the beam center.
Now consider the situation when the particle is located on the beam axis slightly
ahead of the focal point, as is shown in Fig.11.3a. In this case the forces due to the
refraction induced momentum changes are radially symmetric on the sphere and the
net gradient force points downstream in the direction of the photonflow. This
causes the particle to be displaced slightly downstream from the exact position of
the beam waist. In addition, the scattering force also points downstream. Once the
particle is pushed past the focal point, as is shown in Fig.11.3b, the gradient force
now points upstream and eventually counterbalances the downstream scattering
force. Thereby, the net result of these forces is that a stable three-dimensional
trapping position is achieved for the confinement of dielectric particles.
In general, if the particle is displaced laterally from the focus of the laser beam, it
will experience a restoring force that is proportional to the distance x between the
center of the sphere and the focus of the laser (see Fig.11.2). This force can be
described by the equation

F¼kx ð 11 : 1 Þ

This equation is analogous to the behavior of a small spring obeying Hooke’s
law. Here k is a constant, which is referred to as thetrap stiffness. The trap stiffness
can vary widely depending on the particular design of the optical tweezers and the

F 1 F 2

Fnet

Ray 1 Ray 2

Focal

point

F 2 F 1

Fnet

Ray 1 Ray 2

Fscat

Focal

point

(a) (b)

Fscat

Fig. 11.3 Forces on an axially positioned particle locatedaahead of the focal point andbafter
the focal point

11.1 Optical Manipulation 327