Biomedical Engineering Reference
In-Depth Information
of relative phase and will not generally be impacted. Although the overall intensity drops quite rapidly
with distance from the optical axis, the
z
-component of the fundamental field can be as much as 2/3 of the
x
-polarized component about two beam waists from the optical axis. Depending on the location of the
SHG source within the focal volume, these
z-
polarized contributions may be relatively important, allow-
ing access to tensor elements forbidden by the paraxial approximation.
1.4 extension beyond the thin Sample Limit
In all the preceding sections, the formalism has been limited intentionally to thin or point SHG sources.
Extension to sources that are comparable to or greater than the wavelength of light in dimension requires
revisiting some of the simplifying approximations related to both the local field corrections and the influ-
ence of the electric fields upon tight focusing. Using a Green's function approach, in which each location
within the focal volume serves as a driven radiation source,
11,14
mathematical models have been gener-
ated for describing the predicted net radiation patterns expected for sources of different size, shape, and
position relative to the focal volume, as described Section 1.3.3. In effect, the net emission arises from the
collective interference between all the oscillating source polarizations within the excitation volume.
However, a basic understanding of the key interactions driving many of the interesting interference
effects in SHG microscopy can arguably be most easily illustrated by first considering a Gaussian beam
within the paraxial approximation. For simplicity, it will be assumed that the sample is uniform and semi-
infinite (i.e., with a thickness significantly greater than
z
0
and a width greater than
W
0
). In this limit,
coherences that dictate the measured SHG responses are driven primarily by two key interactions: (i) the
coherence length dictated by dispersion
l
c
and (ii) the coherence length dictated by the Guoy phase shift
l
c
G
.
1.4.1 Phase considerations from Dispersion
he forward coherence length
l
c
f
depends on the presence of dispersion within the sample and is largely
independent of focusing. Considering a plane wave propagating through a semi-infinite medium as
illustrated in Figure 1.5, the difference in refractive index between the two wavelengths results in a
phase-walk between the SHG generated previously and the next infinitesimal contribution generated by
another slice. The phase-walk results in a transition from constructive to destructive interference, and
oscillations in the net SHG intensity, referred to in bulk crystals as Maker fringes.
15
The expression for
the forward coherence length is given by considering the distance required for a phase shift of π between
the SHG and the square of the fundamental beam. This forward coherence typically ranges from 5 to
15 μm in tissues, and 30 μm in water.
16
λ
1
l
c
f
=
⋅
(1.21)
4
n
2
ω
−
n
ω
For an SHG beam propagating in the backward (epi) direction, the phase walk between the funda-
mental and the SHG occurs much more quickly, over distances much less than the wavelength of light.
In this case, the backwards coherence length is given by the following expression:
λ
1
l
b
=
⋅
(1.22)
c
4
n
2
ω
+
n
ω
In the absence of birefringence, all tensor elements are simply rescaled by the same factor in both
the forward and backward directions, such that the coherence length from dispersion affects only the
intensity and not the polarization state of the detected SHG.
