Optical Tweezers - The Nanobiotechnology - page 441

2

∇

2 π m

m

−
+

1
2

3

F

grad =

r I . (19.3)

2

c

.

where
c .is.the.speed.of.light.in.vacuum
m .is.the.ratio.of.the.refraction.index.of.the.particle.( n p ),.and.to.that.of.the.medium.( n m )
λ.is.the.wavelength.of.the.incident.laser

The.light.intensity.of.the.incident.Gaussian.beam,. I ,.is.given.by

1
2

2 2

− x /( 2 σ )

I x ( ) = I e . (19.4)

0

σ π

.

where
I 0 .is.the.intensity.of.the.incident.light
σ.is.the.scattering.cross.section.of.the.spherical.particle

The.scattering.force.is.proportional.to.the.light.intensity.( F scat .∞. I ).and.in.the.direction.of.
the. propagation. of. the. incident. light.. The. gradient. force. is. proportional. to. the. intensity.
gradient.(i.e.,. F grad .∞.∇ I ).and.points.toward.the.gradient.when. m .>.1..The.optical.forces.on.a.
polystyrene.sphere.are.shown.in.Figure.19.2a.

×10 -44

3

0.35

0.45

Laguerre-Gauss
trap

Gaussian trap 2.5

0.25 0.35

0.25
0.15
0.05
-0.05
-0.15
-0.25

2

0.15

1.5

0.05

1

-0.05

0.5
0 -1

-0.15
-8 -0.5 0

Position (m)

0.5 1 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8

z(xλ) z(xλ)

×10 -6

×10 -21

5
4
3
2
1
0

0.5 0.25

Gaussian trap

Laguerre-Gauss
trap

0.2

0.4

0.3 0.15

0.2 0.1

0.1 0.05

0 0

-1
-2
-3
-4
-5

-0.05

-0.1

-0.1

-0.2

-0.15

-0.3

-0.4 -0.2

-0.5 -0.25

-1 -0.5 0 0.5 1 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4

(a) (b) (c)

×10 -6

Position (m) r (xλ) r (xλ)

Figure 19.2
Magnitudes. of. optical. forces. on. a. dielectric. particle.. (a). the. gradient. force. and. scattering. force. on. a. 5.nm.
dielectric. particle. (in. the. Raleigh. scattering. regime),. (b. and. c). the. axial. ( z . direction). and. transverse.
( r . direction). trapping. eficiency. ( Q ). of. a. 4.0.μm. dielectric. particle. in. a. Gaussian. beam. (b). and. non-Gaussian.
beam. (c),. where. Q .=. F /( n m I / c ).. For. all. cases,. the. dielectric. particle. has. a. relative. refractive. index. ( n ). equal. to.
1.59.and.the.surrounding.medium.is.water.( n .=.1.33).and.the.trapping.laser.has.a.wavelength.of.1064.nm..In.
(b. and. c). assuming. NA.=.1.02.. (Panels. (b). and. (c):. Reproduced. with. permission. from. Nieminen,. T.A.. et. al.,.
J. Opt .. A. ,.9,.S196,.2007.)