Biomedical Engineering Reference
In-Depth Information
or smaller fibrils packed closely together, while taking into account the regularity of the packing in the
axial direction. For a given effective phasematching condition (Δ
k
), the incremental conversion ampli-
tude along the propagation direction is approximated by Equation 6.7 assuming a homogeneous domain
of length
L
. Owing to the randomness associated with biological tissues, we cannot measure actual Δ
k
values as they will not be single valued. However, as an example, we can utilize multiples of the mismatch
due to dispersion (Δ
k
1
) to investigate trends associated with higher mismatch. Figure 6.2b illustrates
the normalized buildup of the second harmonic with a phase mismatch of Δ
k
1
(for a Δ
n
= 0.02, which
is approximately the case for collagen over the visible-NIR wavelength range) over the course of one
normalized coherence length
L
c1
= 2π/Δ
k
1
. Analogous curves are also given for the distributive buildup
of higher mismatched conditions given by
m
Δ
k
(
m
is an integer) for multiples of Δ
k
1
. We note that maxi-
mal SHG conversion efficiency will occur for small Δ
k
values and interaction lengths
L
(or domains) on
the order of
L
c
. For illustrative purposes, the respective coherence lengths for
m
= 1 and 2 are depicted
parallel to the
x
axis. We see that for each phasematching condition, the conversion scale is proportional
to sin (
m
Δ
k L
/2) and is therefore domain length dependent, reaching its first maximum at the respective
L
c
, which is normalized by
L
c1
in Figure 6.2b. If the propagation lengths exceed the respective coherence
lengths, the amplitude oscillates sinusoidally (as depicted by the curves associated with larger Δ
k
values).
This suggests that for domains on the order of the coherence length, fields supported by relatively small
Δ
k
values (i.e., regular structures) will dominate, while fields associated with larger Δ
k
values will be
characterized by less-efficient SHG conversion at their corresponding maximum value. In sum, we asso-
ciate large Δ
k
values with lower SHG conversion efficiency. Next, we will continue this analysis by show-
ing that large Δ
k
values support backward SHG emission through relaxed phasematching conditions.
6.3.3 Relaxed Phasematching conditions and SHG Directionality
Here, we consider respective phasematching conditions for forward and backward SHG and how these
relate to fibrillar domains in collagenous tissues. As pointed out by Mertz, backward emission arises when
the SHG-producing assembly provides axial momentum,
K
, which alters the direction of the created pho-
ton [33]. We stress here that this is specific to the SHG creation step and is unrelated to subsequent multiple
scattering of the generated signal in tissue. Owing to the fibrillar hierarchy of collagen (often described
as polycrystalline in nature) and measured dispersion (Δ
n
=
n
(2ω) −
n
(ω) = 0.02), we assume that such
Δ
k
values will exist that the coherence length of the created SHG is on the order of the interfibrillar spac-
ing, thus allowing for the possibility of QPM. This then results in the following relaxed phase conditions:
∆
k
=
K
−
(
k
−
2
k
)
(6.8)
f
f
2
ω
ω
and
∆
k
=
K
−
(
k
+
2
k
)
(6.9)
ω
b
b
2
ω
where Δ
k
f
and Δ
k
b
are the phase mismatches for the forward and backward SHG creation, respectively,
and
K
b
and
K
f
are the respective axial momentum contributions to the backward and forward SHG
creation. These equations are identical to those given by Canalias [35] used to describe periodically poled
crystals. However, here, we do not associate
K
with a single grating wave vector but rather an assembly
of values (due to inherent randomness of collagenous tissues) provided by the medium. Backward SHG
creation implies that in terms of magnitude
K
b
>
K
f
and therefore Δ
k
b
> Δ
k
f
., resulting in a distribution
of “lower” efficiency SHG components making up the overall
B
SHG
. Consequently, shorter coherence
lengths are associated with this component. These equations can be modified to account for focused
initial radiation by replacing
k
ω
by ξ
k
ω
, where ξ is the effective reduction in axial propagation vector due
to the Gouy phase shift [37]. The description that follows later holds for the case where the axial spread of
