Biomedical Engineering Reference
In-Depth Information
(a)
(b)
Time
Energy
FIgurE 1.2
Diagram of an anharmonic oscillator potential energy well of a one-dimensional molecule (a,
solid) with the harmonic oscillator overlaid for comparison (a, dashed) and the time-dependent trace of an oscil-
lation within the well (b, solid). The oscillation can be largely recovered from a weighted sum of several harmonics
(b, dashed, and dash-dotted).
1.1.2 nonlinear Polarizability at the Molecular Level
Nonlinear optical interactions can be easily built into this same conceptual framework by consideration
of an anharmonic oscillator.
2
From inspection of Figure 1.2 for a hypothetical one-dimensional molecule,
introduction of anharmonicity in the potential surface will result in a temporal distortion in the induced
polarization. For moderate driving fields, the distortion will be minimal, but will increase in significance
with increases in the driving field and/or the degree of anharmonicity in the potential energy surface.
These distortions in the time domain are recovered in the frequency-domain by including contribu-
tions from higher harmonics of the driving frequency. This effect is illustrated in Figure 1.2, where the
addition of contributions at the doubled frequency recovers much of the distortion introduced by the
depicted anharmonicity. From the Taylor-series expansion of the potential surface, the magnitude of
the contribution at the doubled frequency scales with the square of the driving field.2
2
Unlike the linear
polarizability, the phase of the frequency-doubled contribution inverts if the orientation of the molecule
flips. Consequently, the net polarization at the doubled frequency (and all higher even harmonics) sums
to zero from an assembly of one-dimensional molecules with equal probability for both orientations.
1.2 Mathematical Formalism
1.2.1 the Molecular tensor in the Local Frame
The molecular description of SHG becomes a bit more interesting for realistic molecules occupying
three-dimensional space. The molecular nonlinear optical properties are described by a 3 × 3 × 3 ten-
sor,
β
(2)
, the magnitude of each
β
ijk
element describing the efficiency of generating
i
i-polarized SHG for
driving fields polarized along the
j
and
k
molecular axes. Fortunately, symmetry properties often reduce
the number of parameters required to describe the molecular tensor well below the 27 possible. In this
chapter, only systems of local uniaxial symmetry (e.g., membranes and fibers) will be considered for two
key reasons: (i) they are among the most common and important classes of systems currently studied
by SHG microscopy (confirmed by their ubiquitous appearance in other chapters), and (ii) they are sig-
nificantly more concise to treat mathematically than the more general case of molecular assemblies of
lower local symmetry.
