Biomedical Engineering Reference
In-Depth Information
function of
l
(sinc function) with a characteristic period of π/Δ
k
: the signal starts to decrease when the
interaction length gets larger than this period.
This length is referred to as the
coherence length
(noted
l
c
) of this particular nonlinear interaction.
π
l
FTHG
=
(3.7)
c
|
k
(
3
ω
)
−
3
k
(
ω
) |
The case of backward (i.e., counter propagating) radiation is described by the same equation, except
that we now have:
E z t
( , )
=
A e
i k z
(
−
ω
t
)
(3.8)
ω
ω
1
E
( , )
z t
=
A z e
i
( )
(
−
k
z
−
3
ω
t
)
(3.9)
3
ω
3
ω
3
We can perform the same analysis as in the case of forward propagation, except with
Δ
k
BTHG
=
k
(3ω) + 3
k
(ω). The coherence length for backward THG is therefore defined as
π
λ
l
c
BTHG
=
≈
(3.10)
|
k
(
3
ω
)
+
3
k
(
ω
) |
12
n
ω
where the approximation holds when neglecting dispersion. With typical biological materials and wave-
lengths,
l
c
FTHG
≈ 10 µm
and
l
c
BTHG
≈ 80 nm
, so that the backward coherence length is much smaller than
the forward coherence length. This difference in direction between the wave vectors is responsible for the
small amount of backward-emitted signals in harmonic generation microscopy, as we will confirm later.
This simple 1D geometry illustrates the importance of the relative phase between the fundamental
and the harmonic field, which results in either constructive or destructive interference. However, in the
case of microscopy, we need to consider a focused excitation beam. We will see that going from plane
waves to focused beams not only changes the intensity distribution (and therefore the effective interac-
tion length), but also changes the phase-matching conditions.
3.2.3 tHG with Gaussian Beams
We refine the previous analysis by now considering a Gaussian excitation beam instead of a plane wave.
Although the Gaussian beam model does not provide an accurate description of the phase and intensity
distributions in the case of high numerical aperture (NA) focusing [13], it usually provides solutions that
are in good qualitative agreement with nonlinear microscopy experiments, at least at moderate NAs.
The most obvious difference with the plane-wave case is the nonconstant intensity distribution. The
main consequence is that the nonlinear interaction is confined within a finite effective interaction
volume. However, the intensity distribution is not the only change: the field distribution near the focal
point illustrated in Figure 3.2 also exhibits a progressive phase slippage along the optical axis
z
known as
the Gouy phase shift [14] that results in an overall π radian phase difference between a plane wave and a
focused beam. The Gouy phase shift is particularly significant for THG phase matching, and makes the
phase-matching conditions derived for a plane wave invalid in the case of a focused beam.
There is no simple way to fully account for the phase and intensity dependence of THG, which is why
researchers often rely on numerical simulations to study the influence of different parameters. However,
simple models can be derived from the simulations to understand the key phenomena: in the case of
the phase influence, one elegant model was proposed by Cheng et al. [15] in which they consider the
interaction between a focused Gaussian excitation beam and a harmonic plane wave propagating along
