Biomedical Engineering Reference
In-Depth Information
where
n
2
is the refractive index of the surrounding medium,
m
(
=
n
1
/
n
2
) is the relative index of the particle, and
a
is its radius.
Thegradientforce,
F
grad
,actingontheparticlesisrepresentedby
the following equation:
F
grad
=
·∇
∝∇
(
p
)
E
I
,
(10.2)
where
p
(
=
α
E
) is the dipole moment, and
I
is the intensity profile
of light.
The scattering force
F
scatter
is also given as
F
scatter
∝
ε
0
2
(
E
∗
×
B
∗
),
B
+
E
×
(10.3)
ε
where
is the dielectric constant of a vacuum and
B
is the magnetic
field.
10.2.1
Angular Momentum Density
The linear momentum density of the light
P
,givenbythetime
average real part ofthe Poynting vector, is written as
ε
0
=
i
ω
0
ε
0
2
(
E
∗
×
B
+
E
×
B
∗
)
2
(
u
∗
∇
u
−
u
∇
u
∗
)
P
=
s
ε
0
2
∂
|
|
2
u
2
z
+
ω
0
k
ε
0
|
|
−
ω
ϕ
u
,
(10.4)
∂
r
ω
0
is the frequency of
the light, and
k
is the wavenumber;
s
is the spin angular momentum
(1 or
where
u
is the amplitude of the light field,
−
1) for clockwise or anti-clockwise circularly polarized light,
and
z
and
φ
are unit vectors along the
z
- and azimuthal directions,
respectively [14, 15]. Thus, the angular momentum
M
, of the light is
given by
M
=
r
×
P
,
(10.5)
where
r
is aunitvector along the radial direction.
Laguerre-Gaussian modes (see Fig. 10.2), eigenmodes of the
paraxial propagation electromagnetic equation in a cylindrical
coordinate system are typical OV, and they have a doughnut-shaped
spatial profile in the far-field and orbital angular momentum due to
a phase singularity.
