Biomedical Engineering Reference
In-Depth Information
0
l
c
-
(
e
ω
)
2
e
2
ω
γ
z
FIgurE 1.5
Difference in refractive index of
ω
and 2
ω
will give rise to a phase walk between the incident field and
the SHG generated at a point
z
. This phase walk can result in interference between SHG generated in two different
points along
z
.
γ
shows the oscillation between 100% constructive and 100% destructive interference as a function
of distance between the points of generation of 2
ω
.
l
c
is the distance between these points. The frequency of
γ
will be
proportional to the difference (or sum in the epi direction) of refractive indexes.
Although birefringence within the sample can impact both the forward and backward coherence
lengths, the effect is most significant in the forward direction due to the difference appearing in the
denominator. If the presence of birefringence can further reduce the denominator, the efficiency of
SHG can improve dramatically. Quantitatively incorporating the role of birefringence into the most
general forms for the analysis of semi-infinite media is beyond the scope of this work. However,
full analysis is often not required. In many instances, the polarization dependence is dominated by
the effective tensor contribution corresponding to the longest coherence length when measured in
the limit of gentle focusing. For a uniaxial sample such as a collagen fiber, polarization along the
uniaxial (fiber) axis is defined as the extraordinary axis and polarizations orthogonal to the unique
axis are defined as ordinary. Under normal dispersion conditions, the polarization combination with
the greatest coherence length will typically arise with the fundamental polarized along the unique
(extraordinary) axis and the SHG polarized perpendicular to it (positive birefringence), or vice versa
(negative birefringence). The latter leads to enhancement of the
β
zxx
tensor element, such that it can
often be reasonably assumed to dominate the measured polarization-dependent response. Formally,
this case corresponds to Type I phase matching. In the former case (positive birefringence), the
β
xzz
tensor element would be optimally enhanced, but is symmetry forbidden in uniaxial systems. The
β
xxz
=
β
xzx
tensor elements provide the closest match and experience the greatest enhancement, corre-
sponding to Type II phase matching. If the birefringence is sufficiently high that the coherence length
for the Type I or Type II cases approaches or exceeds the Rayleigh length
z
0
, the one unique “phase-
matched” tensor element can be reasonably assumed to dominate the polarization dependence of
SHG. However, even under ideal phase-matching conditions, which never occur in tissues due to
dispersion and randomness inherent in biological samples,
17
the dominance of a single element can
be tempered significantly as the NA increases due to effective reduction of the net coherence length
from the Guoy phase shift.
1.4.2 the Guoy Phase Shift
Unlike the coherence length contributions described by dispersion, the contributions from the Guoy
phase shift are specific to focused beams such as those employed in microscopy measurements. Because
the image inverts upon passing through the focal volume, the sign for each of the far-field polarization
