Biomedical Engineering Reference
In-Depth Information
Locally uniaxial systems include most naturally formed biopolymer assemblies studied in SHG
microscopy measurements as well as lipid bilayers. In virtually all cases studying fibrous tissues, mea-
surements are performed on assemblies of many fibrils (e.g., collagen, myosin, tubulin, cellulose, etc.),
such that the ensemble fiber retains local uniaxial symmetry, even if the symmetry of the individual
fibrils is lower as in the case of collagen and cellulose. In uniaxial systems, only four unique tensor
elements are required to completely describe the polarization-dependent nonlinear optical properties
within the local molecular-scale frame:
β
zzz
,
β
zxx
=
β
zyy
,
β
xzx
=
β
yzy
=
β
xxz
=
β
yyz
, and
β
xyz
=
β
xzy
= −
β
yxz
=
−
β
yzx
, with the
z
-axis defined as the uniaxial axis.
3
1.2.2 experimental observables and the Jones tensor
To bridge the local tensor to the laboratory-frame measurements, it is useful to first consider the set of
tensor elements accessible within the laboratory frame. Ultimately, the experimental observables will be
dependent on both the intensity and polarization state of the exiting light. A concise means of represent-
ing the polarization state of coherent light is through a Jones vector, which is a unit vector describing the
relative amplitude and phase of, in this case, the horizontal and vertical optical fields (indicated by the
subscripts
H
and
V
, respectively).
=
e
e
ω
e
H
ω
(1.2)
L
ω
V
L
The elements within the Jones vector can be real in the case of linearly polarized light or complex
valued in the more general case of elliptical or circularly polarized light. The use of Jones vector repre-
sentations is limited to characterizing light with a well-defined polarization state. In turbid matrices,
or for partially depolarized light, Stokes vectors and Mueller matrices should be used instead,
4
which is
beyond the scope of this work.
Every optical element that impacts the amplitude or polarization state of the light can be described
mathematically by a corresponding Jones transfer matrix.
4,5
By matrix multiplication, all polarization-
dependent observables can be uniquely predicted from precise knowledge of the initial Jones vector,
provided the subsequent Jones transfer matrices are also known.
By extension to second-order nonlinear optics, an analogous Jones tensor can be generated.
6
This
phenomenological 8-element 2 × 2 × 2 tensor, 6 of which are unique in SHG, is effectively a nonlinear
polarization transfer tensor, completely describing the expected polarization state generated experi-
mentally for any combination of incident polarization states, both of which are typically coming from
the same source in SHG. It serves an analogous role as the Jones matrix for reducing down the collective
influence of a complex optical path to a single polarization transfer matrix. Since all possible experimen-
tal outcomes in an SHG measurement can be determined from detailed knowledge of the Jones tensor,
it also contains all the information experimentally accessible in a single measurement configuration.
Mathematically, the link between the Jones vector of the incident light and the corresponding Jones
vector of the doubled light can be written in terms of a matrix product.
7
( )
2
χ
χ
χ
χ
χ
χ
J HHH
,
( )
2
J HHV
=
( )
2
(
e
ω
)
2
2
e e
ω ω
(
e
ω
)
2
0
0
0
e
H
H V
V
J HVV
,
2
ω
⋅
(1.3)
L
0
0
0
(
e
ω
)
2
2
e e
ω ω
(
e
ω
)
2
( )
2
H
H V
V
J VHH
,
L
( )
2
J VH
,
V
( )
2
J VVV
,
L
