Biomedical Engineering Reference
In-Depth Information
it is in principle fairly straightforward to determine the most probable angles
θ
and
ϕ
for a given set of
polarization-dependent measurements. Similarly, if one knows the orientation angles
a priori
, measure-
ments of the laboratory-frame tensor elements can enable determination of the local tensor.
1.3 Special considerations
To a first-order approximation, one might initially assume that the polarization state of the fundamental
light within the focal volume will simply be equal to the polarization state of the incident light in the
far field. However, this assumption neglects two key interactions: (i) the perturbations due to local field
factors, which correct for the screening arising from the local dielectric medium in which the nonlinear
optical source is embedded, and (ii) the influence of focusing on the local electric fields within the con-
focal volume. Both are considered here.
1.3.1 corrections for the Local Dielectric Medium
For a point source embedded in a dielectric medium, the local fields experienced by the SHG-active
source and detected from that source will be affected by the local dielectric properties. Introducing cor-
rection terms accounts for these perturbations. In the limit of a small source relative to the wavelength
of light, the corrections are quite straightforward to apply, generally affecting only the relative magni-
tudes of the different polarization components. Lorentz corrections are arguably the simplest analytical
expressions routinely used for handling such screening effects. The influence of the local field correc-
tions can be introduced by rescaling each element.
( )
2
( )
2
β
=
L L L
2
ω ω ω
β
(1.11)
.
ijk eff
ii
jj
kk
ijk
The Lorentz scaling factors can be written as elements of diagonal matrices of the following form:
n
2
+
2
0
0
x
1
3
L
ω
=
0
n
2
+
2
0
(1.12)
y
0
0
n
2
+
2
z
There are some lingering questions about the appropriate form for the local field correction matrix
describing the doubled light
L
2ω,10
but the most common approach in practice is arguably to use the
same expression, but with the optical constants evaluated at the doubled frequency. The primary
goal of this work is to define and describe a general framework, and we leave it up to the reader to
decide the most appropriate specific model to use to treat the local field factors for the exiting beam
should the use of the effective tensor prove insufficient for the analysis. In a uniaxial system (e.g., a
thin fiber) with the
z-
axis defined as the unique axis, the local dielectric constant will generally be
anisotropic, with
n
x
=
n
y
=
n
o
and
n
z
=
n
e
(i.e., the ordinary and extraordinary refractive indices of the
fiber, respectively).
Most generally, the influence of the local field corrections can be incorporated mathematically by
multiplication of the 27 element
β
(2)
vector by a 27 × 27 matrix generated from the double Kronecker
produce of the three
L
matrices.
(1.13)
L
=
L
2
ω
⊗
L
ω
⊗
L
ω
