Biomedical Engineering Reference
In-Depth Information
components also inverts. However, the SHG response scales with the square of the fundamental field,
which does not invert. Consequently, the Guoy phase shift results in a net 180° phase walk between the
doubled light and the square of the fundamental beam. This sign inversion arises smoothly across the focal
volume manifest as a phase shift described by the
e
1
0
term derived from Equation 1.15. The Guoy
phase term is summed in the exponent, such that it can be treated additively to other phase-shift contribu-
tions, including those arising from dispersion. The Guoy coherence length is given by the following expres-
sion derived assuming a Gaussian exiting SHG beam with a beam waist in the focal plane matched to the
square of the fundamental beam waist.
−
tan ( /
−
z z
)
=
π
l
c
G
=
2
z
0
tan
∞
(1.23)
2
Because the Guoy phase shift scales as a tangent function, it only approaches, but does not reach, a
total phase shift of
π
. Therefore, by the conventional definition, the coherence length is ∞ for SHG (this
is not the case for THG) but the span over which the Guoy phase shift arises is impacted by the degree of
focusing. Therefore, we introduce the half-coherence length, corresponding to the distance producing
a phase change of 90°.
π
=
l
c
G
/
2
=
2
z
tan
2
z
(1.24)
0
4
0
Because the Guoy phase length depends on
z
0
, higher NA objectives produce smaller values of
l
c
G
/
2
In
the backward direction (epi), the coherence length given by Equation 1.22 can be reasonably expected
to be much shorter than the contributions from the Guoy phase shift and dominate. However, in the
transmission direction if
l
c
G
/2
is significantly smaller than the forward coherence length, it can poten-
tially dominate the net coherence length. Since the Guoy phase term affects each tensor element equally,
it results only in a rescaling of each tensor element in the limit of comparatively long coherence lengths
from dispersion.
The net coherence length of the sample in the forward direction is given by the sum of the contri-
butions from normal dispersion and the Guoy phase shift. Over distances of
l
c
G
/2
spanning the focal
volume, the Guoy phase shift varies approximately linearly with distance, just as
l
c
f
and
l
c
b
. In the limit
of low NA, the Guoy half-coherence length approaches infinity and the dispersive/birefringent terms
dominate. Conversely, under relatively tight focusing conditions and in the limit of weak dispersion/
birefringence, the Guoy half-coherence length can be expected to dominate, impacting the magnitude
and phase of each tensor element equally. In this latter limit, the rescaled
β
ef
( 2
tensor is recovered for
describing the net polarization-dependent SHG properties of the sample. Because the Guoy phase shift
often brings the system close to destructive interference when focusing by inducing an initial shift of
π
,
its presence can make the measurements much more susceptible to perturbations from even relatively
weak dispersion, which can push the system to destructive interference more quickly than for plane
waves. Some knowledge of the optical constants of the sample together with the beam path and the S/N
of the measurements can be helpful for deciding whether one or both contributions need to be explicitly
considered when interpreting image contrast and polarization-dependent measurements.
Finally, it should be noted once more that this analysis fundamentally relies on the validity of the
paraxial approximation, which may not hold quantitatively when using high NA objectives. Under
such conditions, the paraxial approximation provides only a qualitative means for aiding in the inter-
pretation of image contrast, but should be used cautiously for quantitative analysis. The limits within
which the paraxial approximation can still be used for describing polarization-dependent effects in
.
