Biomedical Engineering Reference
In-Depth Information
This equation can easily be integrated, and we find
z
e
i k
∆
ζ
d
i b
ζ
∫
B z
( )
=
B
n
0
(3.34)
(
1
+
2
ζ
/
)
(
n
−
1
)
1
z
0
=
B J
(
∆
k z z
,
, )
(3.35)
0
n
0
where
J
n
is called the
J
-Integral of the
n
-th order process.
In the case of THG, we have
z
e
i k
∆
ζ
ζ
ζ
/
d
i b
∫
B z
( )
=
B
3
0
(3.36)
(
1
+
2
)
2
1
z
0
=
B J
3
(
∆
k z z
,
, )
(3.37)
0
0
This integral unfortunately cannot be integrated analytically in most cases, except in the case of an
infinite homogeneous medium (
z
0
= −∞
, z
= ∞), in which case it yields [43]:
(
n
−
2
)
b
2
π
b k
∆
J
( )
n
(
∆
k
>
0
)
=
e
−
b k
∆
/
2
(3.38)
H
2
(
n
−
2
)
2
( )
(
(3.39)
J
∆ ≤
k
0
)
=
0
H
Despite the simplifying hypotheses, several important properties of THG with focused beams are
described by this integral. In particular, it is found analytically that no THG is obtained from a homo-
geneous, normally dispersive or even nondispersive medium. In addition, the numerical integration of
this integral in the case of slab and interface geometries yields results which are consistent with non-
paraxial simulations.
However, the two main hypotheses do not hold in the case of nonaxially symmetric samples, and a
more complex model must be used.
Green's Function Formalism for tHG
The following model, which is used to represent a coherent nonlinear microscope, has been proposed
by Cheng et al. [15,55]. Figure 3.20 illustrates the geometry and the notations used. It consists of three
elements:
1. The excitation field near focus is calculated from the field distribution at the back aperture and
from the objective properties.
2. The interaction between the fundamental field and the sample is described by the spatial distribu-
tion of a nonlinear tensor (the 4th rank χ
(3)
tensor in the case of THG).
3. The propagation of the nonlinear polarization to the far field is calculated using Green's functions.
This model neglects the spectral width of the excitation pulses: only the central frequency is consid-
ered. Therefore, all effects related to the spectral phase of the excitation are neglected. This approxima-
tion is justified by the fact that chromatic effects are negligible over the extent of the excitation volume
for the pulses typically involved in THG microscopy (100 fs duration, 5−10 nm bandwidth) in the
