Biomedical Engineering Reference
In-Depth Information
1.1 Physical overview
1.1.1 Linear Polarizability at the Molecular Level
Before tackling nonlinear optical effects, it arguably makes sense to describe linear optical effects within
the same conceptual framework to better understand the close similarities. At the single molecule level,
the linear polarizability describes the magnitude and phase of the dipole induced in the molecule by an
AC or DC field. For an isolated molecule at optical frequencies, the linear polarizability describes optical
scattering. When the density of scattering centers is high enough such that many are located distances
much less than the optical wavelength, interference between the collective set of individual dipoles leads
to directional reflection and refraction of light.
This linear polarizability can be qualitatively interpreted as the efficiency of “sloshing” the electron
cloud of a molecule in a particular direction when driven by a polarization in a particular direction (not
necessarily the same). Considering first a one-dimensional “molecule” as an illustrative example, the
linear response is depicted in Figure 1.1. For a low applied field,
e
, the polarization,
P
, induced in the
molecule will undergo harmonic oscillation, the magnitude of which is inversely related to the curvature
of the surface. A high curvature corresponds to small charge displacement and induced dipole, and vice
versa. (Although the potential energy surface holding the multi-electron cloud to the molecule is often
not known, Taylor-series expansion about the zero-field energy minimum always produces a harmonic
local potential close to the minimum sampled at low field strengths.) Formally,
P
is related to the driv-
ing field.
e
by matrix multiplication, each element of which is inversely proportional to the curvature of
the potential energy surface as in Equation 1.1. For a three-dimensional molecule, the polarizability is
described by a 3×3 matrix,
α
, each
ij
element of which describes the magnitude and phase of the
i
-polarized
dipole generated for a
j
-polarized driving field. Matrix multiplication of the polarizability matrix by the
incident field describes the direction and magnitude of the induced polarization.
1
P
P
P
α
α
α
e
e
x
xx
xy
xz
x
=
α
α
α
⋅
(1.1)
y
yx
yy
yz
y
α
α
α
e
z
zx
zy
zz
z
(a)
(b)
Time
Energy
FIgurE 1.1
Diagram of the potential energy well of a one-dimensional molecule under the harmonic oscillator
approximation (a) and the time-dependent trace of an oscillation within the well (b). This approximation is valid in
the limit of low-energy driving fields.
