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3.2 Theoretical Analysis 93
Finally, Q s and Q g are obtained by integrating all the rays using

2π R m R m

1 2
Q s = rQ sz (r)drdβ = rQ sz (r)dr,
πR 2 m R 2 m
0 0 0
2π R m R m

1 2
Q g = rQ gz (r)drdβ = rQ gz (r)dr.
πR 2 m R 2 m
0 0 0

2
2
The total trappingefficiency is given by Q t = Q + Q .
g
s
3.2.3 Effect of Beam Waist
In the ray optics, a laser beam is decomposed into individual rays with appro-
priate intensity, direction and polarization, which propagate in straight lines.
In actual conditions, the focused light beam has a beam waist, which means
that each ray varies its direction near the focus. Therefore, the incident angle
θ 1 varies from that of the straight line, leading to the recalculation of the
exact optical pressure force.
We introduce a Gaussian beam profile (3.9) of a beam waist ω 0 and the
depth of focus Z 0 instead of straight line ray optics as

λ 2
ω 0 = ,Z 0 = kω , (3.9)
0
2NA
where k is the wave number 2π/λ, λ is the wavelength, and NA is the numer-
ical aperture of the objective.
To determine the incident angle θ 1 (r) of a Gaussian ray passingat r = r in
the aperture of the objective enters at the point (α, β) on the sphere surface
as shown in Fig. 3.14. The coordinates (α, β) are expressed

2 2

2 2 2 2 2 2 r 2 2 r 2
2sZ − 4s Z − 4Z 0 s − r + R m ω 0 Z + R m ω 0
0
0
0
0
α = ,
2
2
2 Z + r ω 2
0 R m 0
(3.10)

2
2
β = r − (s − α) . (3.11)
0
Then the incident angle θ 1 (r) is calculated as the angle between the tan-
gent vector a of the Gaussian ray at (α, β) and the direction vector b pointing
to the center of the sphere. After the incident angle θ 1 (r) is defined, the trap-
pingefficiency alongthe optical axis can be computed. Figure 3.15 show the
result for a polystyrene sphere suspended in water. Consideringthe beam