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3.3 Experimental Measurement and Comparison 111
16
Minimum trapping power (mW) 10 8 6 4 Glass
14
12

0 2 Polystyrene
0 10 20 30 40 50 60 70
Stage velocity (mm/s)

Fig. 3.37. Relationship between minimum trapping power obtained using solitary
optical fiber and stage velocity for microsheres of 10 micrometer in diameter

microspheres. This is because the trappingforce is equivalent to F g − F s
for downward illumination, but to F g + F s for upward illumination. The-
oretically, the minimum trappingpower ratios between the upward- and
downward-directed beams are 1.8 for polystyrene sphere and 1.6 for glass
sphere.
2. The experimental minimum trapping powers are in fairly good agreement
with the theoretical ones for axial trapping, but not in good agreement for
transverse trapping. This is because the trapping position for transverse
trapping changes due to the large gravitational force, particularly for high-
density and/or large particles.
3. The minimum axial trappingpower increases as the trappingposition
increases from the chamber surface. This is because the spherical aberra-
tion due to the refractive index difference between the immersion oil of the
objective and the aqueous medium in which a microsphere is suspended.
4. Brownian motion is active for microspheres less than about 1 µm in diam-
eter, which increases trappingpower.
5. Optical fiber trappingis expected to improve both the operation and
implementation.
Example 3.6. Show that the force due to Brownian motion of a microsphere
suspended in water is equivalent to F =2kT/d where k is the Boltzman con-
stant, T is the absolute temperature and d is the diameter of a microsphere.

Solution. Microspheres smaller than about 1 µm in diameter seem to fall out
of the optical trap when laser power is reduced below a certain level. This
is due to the thermal energy driving the particle in the weakest direction of
the optical trap, i.e., parallel to the beam axis. To express the thermal effect
2
in force units we assume that the harmonic trap potential Kz /2(K is the