Biomedical Engineering Reference
In-Depth Information
to backward creation ratio (
F
SHG
/B
SHG
) based solely on initial emission before scattering. The second-
ary filter effect is then determined by the bulk optical parameters at the SHG wavelength by simu-
lating the transmission,
T
2ω
, and reflection,
R
2ω
. By running the trajectories of 50,000 photons, the
detected forward (
F
) and backward (
B
) components as well as the attenuation are then simulated.
This approach allows for the decomposition of the SHG creation and propagation dynamics, where
experimentally the sources of the photons, that is, from direct emission or arising from multiple
scattering, are indistinguishable.
6.3 SHG Phasematching in tissues
6.3.1 introduction to Phasematching
Under the right conditions, that is, angle of incidence, and polarization or artificially induced peri-
odic grating, uniaxial birefringent crystals such as KDP and BBO can display ideal phase matching
for SHG, that is, Δ
k
=
k
2ω
− 2
k
ω
= 0, where
k
2ω
is the wave vector for SHG photon and
k
ω
is the wave
vector for the incident photon. Accordingly, this condition is also characterized by infinite coher-
ence length,
L
c
= 2π/Δ
k
, and is 100% forward propagating. However, the physical situation in tissues
is different and there is always some extent of phase mismatch, that is, Δ
k
≠ 0, due to the underly-
ing polycrystalline nature of most collagenous tissues [32]. As a consequence of this structure, these
materials contribute axial momentum to the lattice, altering the ideal phasematching conditions
described earlier. Specifically, the fibrillar packing density and randomness alter the conservation of
momentum establishing a quasicoherent process, which results in an emission directionality where
the relative shares of the forward and backward components depend on the extent of the mismatch.
In general, strict phasematching conditions are not applicable in this case, as there are no type I (or
angle-tuned) phasematching conditions, and the minimum mismatch for the forward SHG (
F
SHG
) is
governed by the dispersion between the laser and SHG wavelengths, (
n
2ω
≠
n
ω
). Additionally, as will
be shown later, backward emitted SHG (
B
SHG
) requires strong axial momentum contributions from
the media if the SHG-created photons are to travel in the opposite direction of the incident photons
and still conserve momentum. The SHG conversion efficiency (related to χ
(2)
) is also then determined
by the phase mismatch, by axial momentum contributions from the media, as well as decreased by the
randomness inherent to biological tissues.
Previously, using antenna theory, Mertz predicted that spatial inhomogeneities (axially periodic and
spherically localized distributions) are capable of contributing such momentum to the phasematching con-
dition and, under appropriate conditions, can account for the creation of backward emitted SHG [33]. This
poses a question as to how tissues, in practice, contribute sufficient axial momentum to create
B
SHG
. hrough
our model, we demonstrated that only through quasiphasematching (QPM) can appreciable
B
SHG
be pro-
duced by intensity buildup along multiple fibrils. Moreover, this treatment explains why backward SHG
creation is not appreciably observed in dye-labeled membrane case due to lack of distributive amplification
along its very thin axial extent (~4 nm). As Mertz's treatment only considered a single scattering cluster (i.e.,
individual dye molecules in a membrane) and neglected dispersion and randomness, this theory does not
include the factors necessary for consideration of the SHG directionality and conversion efficiency in tissues.
This situation differs from the limiting membrane case, as axially adjacent fibrils are packed sufficiently close
(on the order of the coherence length) to interact and contribute to the overall SHG response.
6.3.2 Heuristic Model of Phasematching in collagenous tissues
To describe SHG creation in fibrillar collagen, we build upon and expand Mertz's formalism, by defining
relaxed phasematching conditions. We introduce the concept of a domain of SHG-producing structures,
which can be a collection of smaller fibrils, larger fibrils, or fibers packed together in the axial direction
on the size scale of less than λ
SHG
. This generalizes previous efforts based only on fibril size and without
