Biomedical Engineering Reference
In-Depth Information
involved in the articles from which the illustrations were adapted. In addition, we thank Nadine Peyriéras
and Louise Duloquin for preparing the zebrafish embryos, Laurent Combettes for the hepatocytes, and
the French Eye Bank and the ophthalmology department at Hôtel-Dieu Hospital for the supply of human
corneas. This work was supported by Centre National de la Recherche Scientifique (CNRS), Institut
National de la Santé et de la Recherche Médicale (INSERM), Délégation Générale de l'Armement (DGA),
Agence Nationale de la Recherche (ANR), and Fondation Louis D. de l'Institut de France.
Appendix
In this appendix, we summarize the two commonly used formalisms for calculating THG from various
geometries. The first model assumes scalar Gaussian beam excitation, moderate focusing, axially sym-
metric samples, and provides some analytical results. The second model uses a Green's function formal-
ism and is more general, but its implementation is more computation-intensive.
tHG from an Axially Symmetric Sample with a Focused Gaussian Beam
Here, we follow the method used in Boyd's
Nonlinear Optics
[43]:
1. The fundamental (
n
= 1) and
n
-th harmonic beams are described as 2D (ρ,
z
) Gaussian beams:
B
iz b
n
2
2
1 2
A
( , )
ρ
z
=
e
−
r w
/
(
+
iz b
/
)
(3.29)
n
n
n
1
+
2
/
n
where
w
0
is the waist of the fundamental beam,
w
2
=
w n
2
/ ,
b
=
ω
k w
2
denotes the confocal
n
n
n
parameter, and
k
n
ω
= 2π
n
ω/
n
n
ω
is the wavenumber.
2. We only consider interactions along the
z
-axis (i.e., slabs or interfaces), and the intensity of the
fundamental beam is considered constant.
The equation is modified to account for the
z
dependence of the harmonic signal:
B z
iz b
( )
2
2
1 2
A r z
( , )
=
n
e
−
r w
/
(
+
iz b
/
)
(3.30)
n
n
n
1
+
2
/
n
=
B z G r z
n
( )
⋅
( , )
(3.31)
n
with
G r z
2
2
1 2
.
( , )
= +
1
2
iz b e
/
−
r w
/
(
+
iz b
/
)
n
n
n
n
The paraxial wave equation for
n
-th harmonic generation can then be written as
∂
∂
C G r z
( , )
n
=
∇ +
(
B z G r z
( )
⋅
( , ))
2
2
ik
n
1
n
n
r
n
ω
z
∂
∂
∂
B z
z
( )
n
=
B z
( )
∇
2
(
G r z
( ,
))
+
2
ik B z
( )
z
G r z
(
( , ))
+
G r z
( , )
n
r
n
n
ω
n
n
n
∂
∂
∂
(3.32)
=
G r z
( , )
(
B z
(
))
n
n
z
since
G
n
(
r,z
) is a solution to the paraxial wave equation.
Moreover, since
G r z
G r z
( , )
,
n
e
iz b
i kz
∆
1
=
(3.33)
(
1
+
2
/
)
(
n
−
1
)
n
