Biomedical Engineering Reference
In-Depth Information
E
x
E
y
E
z
E
2
×1
×15
×3
FIgurE 3.13
Vectorial aspect of the focused Gaussian beam. Calculated electric field at the focus of a Gaussian
beam with initial
x
-oriented linear polarization. −1 μm <
x,y
< 1 μm. Conditions:
NA
= 1.4,
n
= 1.5.
For simplifying the discussion, we will however neglect this
z
polarization component in the remain-
ing of this chapter. This approximation is valid for THG microscopy applications using moderate excita-
tion NAs. We will therefore consider only plane linear and circular polarization distributions.
In a medium characterized by its third-order nonlinear tensor
χ
ijkl
( )
( )
3
r
, the excitation field induces a
nonlinear polarization described by
=
∑
P
(
3
ω
)
χ
( )
3
E E E
(3.22)
i
ijkl
j
k
l
ij ik il
,
,
In the case of a homogeneous isotropic medium, the χ
(3)
tensor verifies [43]:
(3)
χ
=
χ δ δ
0
(
+
δ δ
+
δ δ
)
(3.23)
ijkl
ij
kl
ik
jl
il
jk
and we can express the nonlinear polarization induced by the exciting field
E
in Cartesian coordinates as
E E
(
2
+
E
2
+
E
2
)
x
x
y
z
P
(
3
ω
)
=
3
χ
E E
(
2
+
E
2
+
E
2
)
(3.24)
0
y
x
y
z
E E
(
2
+
E
2
+
E
2
)
z
x
y
z
Several points can be noticed from this equation:
1. Although THG is a third-order process,
P
y
and
P
z
depend both linearly and nonlinearly on
E
y
and
E
z
. This means that one has to be careful before deciding to neglect the vectorial components of
the exciting field. Of particular importance is the spatial overlap between the different polariza-
tions. For example (see Figure 3.13), if we consider a focused Gaussian beam with initial linear
polarization along
x
, the
E
y
and
E
z
components of the electric field have their maxima away from
the optical axis, and so their overlap with
E
x
is small. This is one of the reasons why the paraxial
scalar approximation works well for THG microscopy.
2. For an isotropic medium, if we consider a circular polarization in the paraxial approximation,
we have
E
x
=
i
.
E
y
so
P
x
=
P
y
= 0, that is, no third-order polarization. For that reason, no THG is
obtained from isotropic media excited with circularly polarized light, even at interfaces [44]. One
consequence is that THG with circular incident polarization can be used to discriminate between
isotropic and anisotropic media. It also has implications in the case of combined THG/SHG imag-
ing: linearly polarized excitation is generally preferred for general-purpose THG imaging, how-
ever, linear polarization results in orientation effects particularly pronounced in SHG images.
3.3.2
χ
(3)
tensorial effects
We have seen that the absence of THG signal using circularly polarized light in isotropic media comes
from the tensor symmetries. We will now consider anisotropic media, and first neglect refractive index
