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4.2 Theoretical Analysis I – Optical Torque 127
4.2.2 Optical Rotor with Slopes on the Light-Incident Surface

The characteristics of the optical trappingforce and optical torque for a cylin-
drical optical rotor with slopes on the light-incident surface are analyzed using
a ray optics model for both parallel and focused laser beam illuminations. The
rotor is expected to be aligned with the light beam propagation axis. Since
the total illuminated light beam contributes to the rotation and the cylindri-
cal shape is effective in decreasingthe viscous dragforce, this new rotor is
expected to rotate much faster than the conventional one.
First, light-driven cylindrical rotors with various slope angles and height-
to-radius ratios are analyzed. Figure 4.4 shows that the optical pressure force
F perpendicular to the surface, at an arbitrary point on the top surface is
torsionally directed alongthe beam axis. Force F is decomposed into two
components: scatteringforce F s pointingin the direction of the beam axis
and gradient force F g pointingin the direction perpendicular to the beam
axis. Gradient force F g (not shown) is decomposed further into torque force
F t and radial force F r . On the lower surface, only scatteringforce F b is exerted,
and no z-axis torque exists because the surface is perpendicular to the optical
axis.

Parallel Beam Illumination
We assume that a circularly polarized Gaussian Nd:YAG laser beam (wave-
length λ =1.064 µm, power P = 100 mW) illuminates the rotor (refractive
index n 2 =1.5, density ρ =2.2gcm −3 , diameter 2r =3 µm and height
h =10 µm) in water (n 1 =1.33). When vertically illuminated on the top
surface by a parallel beam, the incident angle a 1 is equal to a (the slope angle
of the rotor) and the optical pressure F at arbitrary point A is given by (4.2).
Quantities R and T are derived from the Fresnel reflection and transmission
coefficients using(3.2) and (3.3). Scatteringforce F s and torque force F t at
point A are given by

F s = F cos(a) (4.5)
F t = F g sin(θ)= F sin(a)sin(θ) (4.6)
Therefore, torque T q at point A is

T q = rF t = Q torque (n 1 P/c), (4.7)

where Q t is the torque efficiency in unit of m.
When vertically illuminated by a parallel beam, all the refracted light is
reflected by the side surface, which leads to the incident angle to the bottom
surface being(a 1 − a 2 ). Therefore, optical pressure F b at the bottom surface
is given by

F b = {(n 2 /n 1 )(1 + R )cos(a − a 2 ) − T cos(a 3 )}(n 2 P /c), (4.8)