Biomedical Engineering Reference
In-Depth Information
dispersion and linear birefringence (i.e.,
n
ω
=
n
3ω
everywhere). However, the nonlinear tensor consid-
ered is not that of a homogeneous isotropic sample anymore.
3.3.2.1 tensors and nonlinear Polarization
The form of the isotropic tensor has been given in Equation 3.23. The case of the third-order nonlinear
susceptibility for various symmetries has been studied by Butcher [45], expanded by Hellwarth [46], and
can be found in a number of references, including Ref. [43].
For simplicity, here, we restrict our analysis to the case of a crystal with one main axis along the
x
-axis
with a
C
∞v
symmetry. This tensor has three independent tensor elements:
χ
=
=
χ
xxxx
χ
χ
=
χ
=
χ
=
χ
=
χ
xxyy
xyyx
xyxy
xxzz
xzzx
xzxz
χ
=
χ
=
χ
=
χ
=
χ
=
χ
(3.25)
yyzz
yzzy
yz
yz
zzyy
zyzy
zyyz
=
χ
=
χ
=
χ
=
χ
=
χ
=
χ
yyxx
yxxy
yxxz
zzxx
zxzx
zxxz
χ
zzzz
=
χ
=
3
χ
yyyy
yyzz
If we assume χ
||
= χ
xxxx
, χ
cr
= 3χ
xxyy
, χ
⊥
= 3.χ
yyzz
, we can then express the nonlinear polarization as
(still neglecting
E
z
):
E
(
χ
E
2
+
+
χ
E
2
)
x
x
cr
y
P
(
3
ω
)
=
E
(
χ
.
E
2
χ
.
E
2
)
(3.26)
y
cr
x
⊥
y
0
Let us now consider an interface between such a medium and an isotropic medium (“air”) sharing the
same linear properties but with χ
(3)
= 0, and look at the influence of two parameters on the THG signal:
the incoming polarization and the ratio between the tensor elements.
3.3.2.2 interface between an Anisotropic Medium and Air
Isotropic medium, linear polarization
Figure 3.14 illustrates THG signal obtained in the classical situation [8,9,15] where the focus is
z
-scanned across the interface between an isotropic medium (χ
(3)
= χ
0
) and air (χ
(3)
= 0), where
z
denotes the distance between the interface and the focal plane.
(a)
(b)
1.0
0.8
0.6
Exc.
THG
0.4
0.2
z
0.0
-3
-2
-1
0
1
2
3
Interface position (µm)
FIgurE 3.14
THG from an interface with linearly polarized light. (a) Geometry considered. (b)
C
0
(
z
): numerical
calculation of the THG signal obtained when the excitation beam is
z
-scanned across an
xy
interface between an
isotropic medium an air.
NA
= 1.2, λ = 1.18 μm.
