Biomedical Engineering Reference
In-Depth Information
SHG is the coherent version of HRS. In both cases, molecules exposed to the electric field need to possess
a dipole moment, which ensures that a harmonic optical wave component can be produced as a result
of the nonsymmetrical oscillation of the electrons in response to the symmetrically oscillating driving
wave. An additional noncentrosymmetric component can arise from molecular chirality [10-12]. The
efficiency of HRS is characterized by the molecular first hyperpolarizability, β, which relates to the
second-order induced dipole moment of the molecule as follows:
d
( )
2
= β
E
2
(4.2)
The second-order nonlinear polarization P (2) response of a material is simply the bulk representa-
tion of d (2) , and the emitted SHG is a coherent addition of the HRS emission from the local molecular
ensemble. The bulk property χ (2) can be related to the molecular-level property β as below:
χ
( )
2
=
N s
β
(4.3)
where N s is the number of molecules involved for the coherent SHG generation and 〈β〉 is the orienta-
tional average of β. We want to point out that E , P , d (2) are all vectors, whereas β and χ (2) are tensors, and
thus 〈β 〉 is zero for randomly orientated molecules such as in liquid solutions, and this type of mate-
rial cannot produce SHG. HRS emits in all directions, and only when the molecules are packed in an
organized way is 〈β 〉 nonzero, and the frequency-doubled emission becomes predominant in a certain
direction as a result of the collective response of all the molecules to the driving optical field. In sum-
mary, effective SHG requires that the molecules of the medium have a permanent dipole moment and
nonzero hyperpolarizability, and, at the bulk level, the dipole moments be aligned as an organized array.
In a microscope, the bulk level translates to the focal volume.
Even if a medium meets all of the aforementioned three requirements, SHG is typically still weak.
One strategy to increase the SHG intensity is to utilize resonant enhancement. This can be done as illus-
trated in Figure 4.1. SHG can be viewed as a wave mixing process where two input photons are coupled
to form one output photon via a virtual energy level. However, if the SHG excitation energy is resonant
Virtual
S *
1
S 1
ω
ω 1
2 ω
ω 2
ω
ω 1
S *
S 0
0
S 0
SHG
TPEF
FIgurE 4.1 Energy diagram for SHG and TPEF. S 0 is the ground state; S * is the vibrationally excited S 0 state; S 1
is the lowest singlet excited state; S * ( n ≥ 1) is a vibrationally excited lowest or higher singlet excited state. The thick
solid arrows refer to excitation and the thin solid arrows refer to emission. Δ 0 and Δ 1 represent relaxation that leads
to heat dissipation into the solvent.
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