Biomedical Engineering Reference
In-Depth Information
with a real excited state, the SHG intensity will be greatly enhanced. A simple two-state model indicates
[4,13,14] that the hyperpolarizability can be written as
ω
f
∆
µ
ge
ge
ge
(4.4)
β
∼
(
ω
2
−
ω ω
2
)(
2
−
4
ω
2
)
ge
ge
where ω
ge
is the frequency that corresponds to the energy gap between the ground state (
g
) and the upper
state (
e
),
f
ge
the oscillator strength that describes the optical transition probability from state
g
to state
e
, and Δμ
ge
denotes the dipole moment change upon the optical transition. Equation 4.4 clearly indi-
cates that when the excitation energy (2
ħ
ω) matches the energy difference the two states (
ħ
ω
ge
), maximal
hyperpolarizability is achieved, as the denominator goes to zero, which results in enhanced SHG signal.
However, utilizing resonant enhancement will result in photobleaching in-plane. This condition is nec-
essary for SHG imaging of dye-labeled membranes but is not typically employed for imaging proteins.
4.2.1.2 SHG emission Directionality
An additional consequence of the coherent nature of SHG is the emission directionality and under-
scores a large difference relative to the incoherent process of TPEF. Neither single nor multiphoton
excited fluorescence has a phase relationship to the excitation laser and is emitted over 4 pi steradians.
In strong contrast, SHG has a phase relationship with the laser and a well-defined emission directional-
ity. The directionality is explained in terms of phase-matching conditions, a requirement of momentum
conservation of optical waves, in addition to the energy conservation requirement that the second har-
monic has photon energy twice that of the fundamental. The momentum of an optical wave is
ħ
k
, where
k
is the wavevector. The direction of the
k
vector defines the optical wave propagation direction and its
value is given by
k
2
π λ
/ / , with λ, ω, and
c
being wavelength, angular frequency, and speed of
light in vacuum, respectively. For efficient SHG in nonlinear crystals induced by a laser approximated as
a plane wave ω, momentum conservation requires that
k
2ω
= 2
k
ω
, or Δ
k
=
k
2ω
− 2
k
ω
= 0. This mandates
that the second harmonic (2ω) follow the
forward
direction of the fundamental wave (ω) in the limit of
perfect phase matching, as achieved in certain uniaxial birefringent crystals such as potassium dihydro-
gen phosphate (KDP) and β barium boron oxide (BBO).
However, for SHG microscopy with biological tissues, perfect phase matching is never achieved as no
type I phase-matching conditions exist, that is, matching the refractive index of the ordinary wave of the
SHG with the extraordinary wave of the fundamental. Thus, the minimum phase mismatch is the dis-
persion in refractive index between the fundamental and SHG wavelengths. Additionally, the molecules
(e.g., collagen) are not perfectly aligned in a tissue. Thus, the strict phase-matching requirement of tissues
is relaxed and nonzero Δ
k
is allowed, in such a way that the SHG signal varies with it as follows:
=
(
)
=
(
c
)
m kL
∆
I
∝
sin
(4.5)
SHG
2
where
m
is an integer and
L
the coherence length [15,16]. While the SHG conversion efficiency decreases
for nonzero Δ
k
,
backward
SHG signal is also produced with this relaxed phase-matching condition due
to the need to conserve momentum. Through the development of a general model of phase matching, we
showed that smaller and larger values of Δ
k
are associated with primarily forward and backward SHG,
respectively [15].
At the same time, a laser beam focused by a microscope objective can no longer be treated as a plane
wave, and it experiences a swift phase anomaly (Gouy shift) at the focus [17,18]. This reduces the effec-
tive optical momentum along the prorogation direction and further complicates the phase-matching
conditions [19]. The direct effect of this is that the SHG beam profile does not match the cone-shaped
fundamental beam that propagates from the focal point. Instead, the emission is a two-lobe profile so
