Biomedical Engineering Reference
In-Depth Information
the focused spot is comparable or larger to 1/Δ
k
1
, that is, the material coherence length based on disper-
sion. This condition is valid for image acquisition performed at medium NA.
Forward SHG will be dominated by phasematching with smaller Δ
k
f
values for domains on the order
of
L
c
= 2π/Δ
k
f
, corresponding to small
K
f
values. Although the overall forward SHG signal is a sum-
mation of all the Δ
k
terms, the lower Δ
k
terms will dominate due to their relatively high conversion
efficiency (see Figure 6.2b). Thus, we associate forward SHG primarily with Δ
k
1
(i.e., the maximum
coherence length). By contrast, backward SHG is entirely dependent upon axial momentum provided
by the lattice to redirect the created wave. Therefore, for significant (on the order of
F
SHG
) backward SHG
intensity, the domain size should be less than the coherence length of the forward field (corresponding
to the linear region of Figure 6.2b), while the interfibrillar spacing must be on the order of the coherence
length associated with the backward field. Thus, phasematching conditions support the association of
B
SHG
with relatively larger Δ
k
values. We note that randomness increases the distribution of available
K
values contributed by the medium and therefore the distribution of both Δ
k
f
and Δ
k
b
, which effects
the overall distribution of SHG creation. Thus, we cannot specify the coherence lengths, as they are not
single valued but state that the
F
SHG
has an upper bound limited by the material dispersion and as a con-
sequence of Equation 6.8 and 6.9, it is characterized by longer
L
c
than that for
B
SHG
.
For the purposes of comparative computational analysis, we will associate
F
SHG
and Δ
k
f
= Δ
k
1
and
B
SHG
with larger multiples of Δ
k
1
values (assigned values Δ
k
b
=
m
Δ
k
1
, where
m
is an integer and pertains
to effective mismatch within a single domain). Doing so, one can predict the %
F
SHG
as a function of
normalized domain (normalized to
L
c1
= 2π/Δ
k
1
) by dividing the SHG intensity of Δ
k
1
over the sum of
itself and the respective
m
Δ
k
1
term (calculated from Figure 6.2b). Figure 6.3 shows the resulting %
F
SHG
for
m
= 2-4. Utilizing superposition of these curves, the calculation shows that domains with values
close to
L
c
1
will support predominantly forward emission, while shorter domains will produce essen-
tially even distributions. The emission directionality is highly sensitive to both the domain size and
magnitude of Δ
k
, where larger Δ
k
results, in steeper transitions to higher
F
SHG
. Based upon arguments
made earlier, we also attribute increased randomness with higher Δ
k
values (i.e., shorter
L
c
) and lower
conversion efficiency within a single domain. This analysis demonstrates that both domain size and
randomness play an integral part in SHG emission directionality and that considerations based solely
on fibril size do not form a complete description of the process. We will demonstrate this explicitly for
the OI disease model in Section 6.4.1.2.
2Δ
k
1
3Δ
k
1
4Δ
k
1
100
90
80
70
60
50
0.0
0.2
0.4
0.6
0.8
1.0
Domain/
L
c
FIgurE 6.3
Calculated %
F
SHG
as a function of normalized domain size for several phasematching conditions
(multiples of Δ
k
1
). This calculation shows that domains with values close to
L
c1
will support predominantly for-
ward emission, while smaller domains will produce essentially 50-50% forward and backward distributions. (From
Lacomb, R. et al. 2008.
Opt. Commun
. 281:1823-1832. With permission.)
