Page 114 - Optofluidics Fundamentals, Devices, and Applications

Optofluidic Trapping and Transport Using Planar Photonic Devices 95

the scattering, adsorption, and trapping forces exerted on a particle
(see Svoboda and Block [21], and others [22,48,70]) take the form
8π 3 I α ε 5 2
/
2
F = o m (5-8a)
scat c 3 λ 4
2πε I
F = mo Im( α) (5-8b)
abs cλ
n α
F = m ∇ I (5-8c)
trap c 2 o

where α= 3V(ε − ε )/(ε+ 2ε )
m m
V = particle volume
c = speed of light
ε and ε = dielectric constants of the particle and material
p m
I = optical intensity.
o
Equating F and F with Stokes drag Eq. (5-3) we obtain
scat abs
nk I ⎛ k α 2 ⎞
3
U = mm o m + Im( α) (5-9)
ac ⎝ 6π
o 6πη ⎜ ⎟ ⎠
where k = 2πn /λ and is descriptive of the particle transport velocity
m m
in the Rayleigh regime.
For the case of a particle traveling very near the surface, we could
improve the accuracy of Eq. (5-9) by equating the propulsion forces
with Faxen’s law, Eq. (5-4).
However, it is generally difficult to estimate the distance the par-
ticle is above the waveguide. A conservative estimate, however, could
be obtained by assuming the particle was right near the surface, in
which case a = h.

Transport in the a ê k Regime
When the particle size is much larger than the wavelength of light,
the assumption of a uniform electric field is no longer valid and we
generally require a numerical method to obtain an estimate of the
strength of the trapping forces and propulsion velocity. For details
see Yang and Erickson [69]; however, generally speaking the E and H
fields can be computed either through a full solution to Maxwell’s
equations or by solving the time harmonic wave equation via the
finite element method. Once these solutions are obtained, Eq. (5-7)
can be solved to obtain the net electromagnetic force on the particle,
in all three coordinate directions, as a function of the optical power in
the waveguide. The dynamic tracking of particle motion in a fluid is
a relatively complex simulation, and thus to obtain the net drag on a