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128 4 Optical Rotor

where a 3 is the refraction angle for the incident angle of (a − a 2 ),P is the

incident light power at the bottom, R is the reflectivity and T is the trans-

mittance at the bottom. The total trappingforce is given as the total sum

F trap = (F s + F b )dS, (4.9)

and total torque M opt actingon the rotor is given as the sum

n 1
M opt = P Q torque dS. (4.10)
c
Figure 4.7 shows the dependence of trapping force F trap on slope angle
a. We have defined the positive axial trappingforce to be in the +z direc-
tion. The curves show F trap on the top (upper), bottom (lower), and both
surfaces (total). Since all the rays refracted at the top surface are reflected
from the side surface, scatteringforce F b on the flat bottom is always greater
than scatteringforce F s on the top. Thus the net trappingforce (F b − F s ),
always positive, pushes the rotor away, which leads to two-dimensional (2-D)
trapping.
Figure 4.8 shows the dependence of the rotation rate on the slope angle.
Assumingthat the rotor is cylindrical, we can approximate the rotation rate
2
by M opt = M drag (= 4πµr hω), where M opt is the optical torque of (4.10), µ
is the medium viscosity (µ = 1 mPa s), r is the radius, h is the height of the
rotor and ω is the angular velocity.

Focused Beam Illumination

A ray tracingmethod consideringthe beam waist is employed to analyze the
optical forces exerted by a focused laser beam. Figure 4.9 shows ray tracing for
the rotor illuminated with a focused beam. An incident ray repeats reflection

60
Lower
40
Trapping force (pN) -20 Total
20
0

-40
Upper
-60
0 20 40 60 80
Slope angle (deg)
Fig. 4.7. Dependence of trapping forces on slope angle