Biomedical Engineering Reference
In-Depth Information
outcome. The full 8 × 27 matrix to bridge the Jones tensor and the local effective tensor is then given by
the Kronecker product of three such inverted matrices, two for the driving fields and a third calculated
for the doubled light.
Γ
( ,
r z ϕ
,
)
=
Γ
2
ω
⊗
Γ
ω
⊗
Γ
ω
(1.19)
The Kronecker products in Equation 1.19 will collectively produce an 8 × 27 matrix for relating all 27 ele-
ments of the effective laboratory frame tensor back to the experimentally measured polarization-dependent
Jones tensor observables. The influences from local orientation (Sections 1.2.3 and 1.2.4) and local field fac-
tors (Section 1.3.1) can be systematically incorporated into this same mathematical framework.
χ
χ
χ
χ
χ
χ
χ
χ
J HHH
,
J HHV
,
J HVH
,
J HVV
,
=
Γ( ,
r z
,
ϕ
)
⋅
R
( , )
θ φ
⋅
β
(1.20)
l eff
,
J VHH
,
J VVH
,
J VHV
,
J VVV
,
L
The corresponding 27 × 27 matrix
R
can be easily generated by taking Kronecker products of the
rotation matrix in Equation 1.9;
R
( , )
R R R
However, the matrix for
G
2ω
(and correspondingly,
Γ
2ω
) has not yet been determined in the preced-
ing analysis. This matrix effectively bridges the emission from the source polarization to the detected
intensity and polarization state following the collection optics. Explicit values for calculating the matrix
elements can be generated based on Green's function propagations of the nonlinear source polarization
integrated over the focal volume and propagation of the source through standard matrices for describ-
ing the collection lens.
13
As in the case of the excitation fields, they are numerically calculable provided
that the source polarization as a function of position is known
a priori
.
The good news is that this collective approach provides a very general toolkit for predicting the antici-
pated far-field experimental observables for a known source of arbitrary shape, arbitrary local tensor,
and arbitrary incident far-field polarization state. However, the problem is not trivial to invert in order
to recover the local source polarization from the detected intensity, since the matrices used are generated
based on numerical integration of a very well-characterized source. In other words, if all the interesting
properties of the sample are known
a priori
, the matrix for
G
2
ω
can be explicitly calculated, but the more
practical problem of relating a relatively small number of experimental observations back to the local
structure is not as trivial beyond the paraxial approximation.
Since
G
ω
and
G
2ω
depend on position within the focal volume, the 8 × 27 matrix
Γ
also changes as a
function of source location. The region immediately adjacent to the focal point will be considered explic-
itly, selected by the nature of its practical significance for SHG imaging. Within one beam waist of the
focal point, both the
z-
polarized and
y-
polarized incident fields experience nodes and correspondingly
low amplitudes. In this region of highest intensity, the local fields remain almost exclusively
x-
polarized
for an
x
-polarized incident field. Consequently, provided the source is both thin and located close to the
focal point, the validity of the plane-wave approximation for describing the polarization is recovered.
There will be some uncertainty in the absolute phase of the exiting beam depending on the axial location
of the sample relative to the focal point due to the Guoy phase shift, but the polarization state is a measure
θ φ
=
( , )
θ φ
⊗
( , )
θ φ
⊗
( , ).
θ φ
