Biomedical Engineering Reference
In-Depth Information
consideration for packing [2,6]. Our model includes contributions from QPM and additional phase mis-
match due to dispersion and randomness, all of which lead to the creation of nonideally phasematched
SHG in both the forward and backward directions. QPM allows the buildup of SHG intensity between
anisotropic domains (here either fibers or assembly of small fibrils) without the need for strict phase-
matching conditions, with maximum effect when the domain size and spacing are on the order of the
coherence length of radiation. For example, QPM theory has been utilized to describe the buildup of SHG
in ferroelectric crystals [34], and has been used in the design and fabrication of efficient backward SHG-
producing periodically poled crystals [35]. We note that while periodically poled crystals utilize periodic
structures designed to maximize conversion efficiencies with single values of Δ
k
, similar, although less
efficient, effects are present in tissues, where the collagen has been described as a nematic liquid crys-
tal. Therefore, it is instructive to associate the high QPM conversion efficiency characteristic of periodic
poled crystals with a completely periodic (hypothetical) collagenous structure, and on the other extreme,
low QPM conversion efficiency consistent with a totally random structure. The physiological case will lie
somewhere in between these limits where the conversion efficiency and emitted directionality are depen-
dent upon the fibrillar diameter, interfibrillar spacing, and randomness of the tissue assembly.
To explicitly examine the impact of the relaxed phasematching conditions including axial contribu-
tions from the media, and a mismatch term Δ
k
arising from dispersion and randomness, we begin by
first considering the simple case of the propagation of a plane wave moving through a nonlinear media
in the direction of its
k
vector. Even though this is not a strictly accurate description of the actual case,
which involves focused excitation (NA = 0.8), we argue that general inferences may be obtained for the
case where both the focused beam length (depth of focus) and the coherence length of the incident laser
are longer than the coherence length of SHG radiation. In our experiment, the coherence length of the
Ti:sapphire laser is ~30 microns, whereas the maximum forward coherence length of collagenous tissues
based on dispersion is ~7 microns.
By following the coupled wave treatment of Munn [36] utilizing the slowly varying field approxima-
tion, the distributed amplification of the second harmonic within a homogeneous region or domain is
given by the following equation:
dE
dz
(
2
ω
)
i
n c
d E
ω
= −
2
(
ω
)exp(
i kz
∆
)
(6.5)
eff
2
ω
where the effective hyperpolarizability coefficient
d
ef
(2ω) is proportional to the second-order bulk sus-
ceptibility χ
2
,
n
2ω
is the index of refraction for the second harmonic wavelength,
c
is the speed of light,
and Δ
k
=
k
2ω
− 2
k
ω
is the magnitude of wave vector mismatch between the incident and second har-
monic waves.
Assuming propagation in the
z
direction and neglecting walk-off [37], the total second harmonic
radiation at length
L
(domain length) is the vector sum of all the intermediate constituents (taking into
consideration their respective phases) from lengths 0 <
z
<
L
, and, utilizing the boundary condition
[
E
(2ω,
z
= 0) = 0], can be expressed by
1
−
exp( (
i
∆
k L
))
E
(
2
ω
,
z
=
L
)
=
κ
E
2
(
ω
)
exp( (
i
2
ω
t
−
k L
))
(6.6)
−
i k
∆
2
ω
eff
/ . When the phase matching term Δ
k
=
k
2ω
− 2
k
ω
is equal to
zero, the maximum conversion efficiency is achieved.
For nonzero Δ
k
's, the second harmonic amplitude
E
2ω
(
z
) at a position
L
along the propagation direc-
tion is given by
with the coupling coefficient
κ ω
=
d
nc
sin(
∆
kL
kL
/
2
)
E
=
κ
E
2
(6.7)
2
ω
ω
∆
/
2
