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98 3 Optical Tweezers
micromachines. The optical fiber implementation of such tweezers is simple
and inexpensive. The apparatus that uses a laser diode and an optical fiber
is particularly simple since no external optics such as a dichromatic mirror, a
beam splitter, and filters are required.
Trappingforces can be resolved into two components: the gradient force
F g , which pulls microspheres in the direction of the stronglight intensity,
and te scatteringforce F s , which pushes microspheres in the direction of light
propagation. If a microsphere is located on the light propagation axis, the
gradient forces cancel out, thereby resultingin pushingthe sphere. Therefore,
two counterpropagating coaxially aligned optical fibers are used to trap the
sphere suspended in water [3.13]. Although the sphere is stabilized axially at
a location where the scatteringforces of the two beams balance each other,
the trappingin the transverse direction is weak. The freedom of operation for
the counterpropagating coaxially aligned optical fibers is poor. In this section,
we theoretically analyze an off-axial microsphere trappingforce [3.14] in three
dimensions in order to trap it with a solitary optical fiber.

Analysis of Off-axial Trapping

Trappingefficiency for a microsphere on an optical axis can be calculated,
from axial symmetry, as shown in Fig. 3.19a, by integrating the optical pres-
sure force due to an individual ray in two dimensions. On the other hand,
calculation in three dimensions is necessary for the off-axial trappingeffi-
ciency because of axial dissymmetry. Figure 3.19b shows that a ray enters at

(a)
Y

F g F s Z
F g F s
Total trapping force

(b) Beam profile
Y
Intersection(x,y,z) Incident
F s angle q
1
F g Z
Sphere center
(0,B,A)
Off-axial distance B
Axial distance A
Fig. 3.19. Geometry for calculating trapping efficiency for a microshere when focus
is located on optical axis (a), and at off-axis (b)