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3.2 Theoretical Analysis 97
Next, the trappingefficiency due to a circular element of radius β is
given as
Q c =2πβQ z .

Finally, this trappingefficiency is integrated over the entire cross-section
of the sphere for all individual rays usingthe Shimpson formula under the
conditions in Table 3.5.
Figure 3.18 shows the axial trapping efficiency dependence on the distance
from the optical fiber end for a polystyrene sphere of radii 2.0 and 2.5 µm.
The laser beam profile is Gaussian and the wavelength is 1.3 µm. It is seen
from the figure that trapping force increases as axial distance increases from
zero to a beam waist of 40 µm, i.e., it increases over the region in which
the fiber lens is focusing, and then begins to decrease monotonically as the
beam diverges beyond the focus. Therefore, we can expect that the optimum
dual fiber lens spacingwill exists at a point where axial trappingefficiency is
changing rapidly (see Sect. 3.3.4).

3.2.4 Off-axial Trapping by Solitary Optical Fiber

In recent years, studies of optical tweezers have been conducted on optical-
fiber tweezers [3.12] to improve their operation in the fields of life science and

Table 3.5. Conditions for analysis of tapered lensed optical fiber trapping efficiency
refractive index
water 1.33
particle 1.59
fiber core 1.446
beam waist in the core (µm)5.0
beam waist distance (µm)49.24
radius of curvature (µm)10
wavelength (µm)1.31
particle radius (µm)2–10

0.008
0.007
Trapping efficiency 0.005 Diameter
0.006
5 mm
0.004
4 mm
0.003
0.002
0.001
0
0 50 100 150 200 250 300 350 400
Distance from fiber end (mm)
Fig. 3.18. Axial trapping efficiency dependence on distance from optical fiber end
of polystyrene sphere