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100 3 Optical Tweezers
√
2
2
left side z (x) (t min )= A − r − x in Fig. 3.20a for both the upper and lower
hemispheres. The integration ends at the tangential point between the ray and
the surface profiles of the upper and the lower hemispheres. The integration
upper lower
end points z (t max ) for the upper hemisphere and z (t max ) for the lower
(x) (x)
hemisphere are given by the solution between two equations shown as

2
2
2

r − x = y (x,z) − B 2 +(z − A) 

. (3.15)

2
1+ z 
ω y = tω 0 
Z 0
Then, Q z and Q z are given as
s(x) g(x)
upper lower
z (x) (t max ) z (x) (t max )
Q z s(x) = Q s(x,z) dz + Q s (x,z) dz, (3.16)
z (x) (t min ) z (x) (t min )
upper lower
z (x) (t max ) z (x) (t max )
Q z = Q g(x,z) dz + dz. (3.17)
g(x) Q g (x,z)
z (x) (t min ) z (x) (t min )
Next, our integration goes along the x-axis. Figure 3.20b shows the top
view, indicatinghow to integrate alongthe x-axis. The trappingefficiencies
Q z and Q z in the yz plane are summed alongthe x-axis in the xz plane.
s(x) g(x)
In this case, the integration starts from x = 0 and ends at x = x(u max ), which
is the tangential point between the ray profile (3.13) and the sphere circle
(3.18) in the xz plane
2
2
x +(z − A) = r 2 


. (3.18)
2

1+ z 
ω x = uω 0 
Z 0
Then, Q all and Q all are given as
s
g
x(u max )

Q all =2 Q z s(x) dx, (3.19)
s
0
x(u max )
Q all =2 Q z g(x) dx. (3.20)
g
0
As a result, the total trappingefficiency comes from (3.7). Followings are the
numerical results for the off-axial trappingin three dimensions.
Off-axial Distance and Microsphere Radius Dependence
In the analysis a circularly polarized laser beam by a laser diode with a 1.3 µm
wavelength, a tapered lensed optical fiber with a curvature of 10 µm, and mi-
crospheres 2–10 µm in radius are used under the conditions listed in Table 3.6.
First, transverse trappingefficiency on the off-axial distance (transverse
offset) is analyzed for a polystyrene sphere of 2.5 µm radius located at different