Biomedical Engineering Reference
In-Depth Information
parameter: if the sample has an appropriate molecular and microscopic structure, constructive interfer-
ence will increase the harmonic signal, whereas an inefficient geometry will result in destructive inter-
ference and weak or null signal.
We will in this first part consider the case of isotropic media, and a linearly polarized excitation (and
therefore harmonic) beam. This approximation allows us to neglect the tensorial properties of THG and
to first derive a simpler scalar description.
3.2.2 tHG with Plane Waves (1D case)
Before discussing microscopy, we first remind a few basic concepts of nonlinear optics. Coupling
between the excitation electric field and the produced harmonic field is described by the nonlinear wave
equation, which in the case of THG can be approximated as
n
2
(
3
ω ω
)(
3
)
2
(
3
ω
)
2
∆
E
(
3
ω
)
+
E
(
3
ω
)
= −
χ
( )
3
(
3
ω ω ω ω
;
;
;
)
E
3
(
ω
)
(3.1)
c
2
ε
c
2
0
To discuss the parameters governing THG efficiency, we first consider the case of a plane wave for-
ward propagating along direction
z
> 0 on the optical axis. The fundamental and harmonic waves can
be described by
E z t
( , )
=
A e
i k z
(
−
ω
t
)
(3.2)
ω
ω
1
E
( , )
z t
=
A z e
i k
( )
(
z
−
3
ω
t
)
(3.3)
3
ω
3
ω
3
with
k
= ( / .
Under the assumption of the slowly varying amplitude approximation, the nonlinear wave equation
can be written as
n
ω
ω
c
ω
∂
∂
(
3
ω
)
2
2
ik
z
E
( )
z
= −
χ
( )
3
(
3
ω ω ω ω
;
;
;
)
A e
i k z
3 3
(
)
(3.4)
ω
3
ω
1
ε
c
2
0
⇓
∂
∂z
A z
(3.5)
( )
∝
A e
i
3
(
∆
kz
)
3
where Δ
k
=
k
(3ω) − 3
k
(ω) = 3(
n
ω
−
n
3ω
)ω/
c
is the phase difference between the propagating funda-
mental and harmonic beams, and is called the
wave vector mismatch
, here governed by the dispersion
(
n
ω
−
n
3ω
) of the refractive index.
We can already extract a qualitative view of the THG process from Equation 3.5: (i) the TH amplitude
is proportional to the cube amplitude of the fundamental, and (ii) it is modulated by the phase mismatch
Δ
k
between the co-propagating beams. This equation can be easily integrated, and the harmonic field
after an interaction distance
l
can be written as
l
∫
E
(
3
ω
= ∝ −
,
z
l
)
A e
3
i kz
∆
dz
(3.6)
0
∝
A l
3
sinc
(
∆
kl
/
2
)
In the case of perfect phase matching (Δ
k
= 0), the THG intensity increases quadratically with
l
. If
there is dispersion (
n
ω
≠
n
3ω
) and therefore phase mismatch, THG exhibits an oscillating behavior as a
