Biomedical Engineering Reference
In-Depth Information
∑
∑
P
( )
2
(
2
ω
)
=
χ ω ω
( )
2
(
,
)
E
(
ω
)
E
(
ω
)
=
χ ω ω
( )
2
(
,
)
E
(
ω
)
E
(
ω
)
i
ijk
j
1
k
2
ikj
k
1
j
2
j k
,
k j
,
∑
( )
2
=
χ ω ω
(
,
)
E
(
ω
)
E
(
ω
)
(5.3)
ijk
j
k
j k
,
Therefore, the susceptibility tensor has the following symmetry:
(5.4)
( )
2
( )
2
χ ω ω
(
,
)
=
χ ω ω
(
,
)
ijk
ikj
With these symmetries, the number of independent elements of the tensor decreases to 18. Hence, the
second-order induced polarization can be written as a function of the components of the tensor
χ
ij
( 2
and
of the electric field
E
i
as follows:
E
E
E
2
x
2
y
P
P
P
( )
2
χ
( )
2
χ
( )
2
χ
( )
2
χ
( )
2
χ
( )
2
χ
( )
2
x
xxx
xyy
xzz
xyz
xxz
xxy
2
z
( )
2
=
χ
( )
2
χ
( )
2
χ
( )
2
χ
( )
2
χ
( )
2
χ
( )
2
⋅
(5.5)
y
yxx
yyy
yzz
yyz
yxz
yxy
2
2
2
E E
E E
E E
y
z
( )
2
χ
(
2
)
χ
( )
2
χ
( )
2
χ
( )
2
χ
( )
2
χ
( )
2
z
zxx
zyy
zzz
zyz
zxz
zxy
x
z
x
y
The susceptibility tensor
χ
ij
( 2
can be expressed in the HRS molecule's system of coordinates (
x
′,
y
′,
z
′)
calculating each tensor component by the following rotation:
=
∑
′
( )
2
( )
2
χ
cos
ϕ
cos
ϕ
cos
ϕ χ
(5.6)
ijk
ii
jj
′
kk
′
i k j
′
′
′
i
′
j k
′
′
where φ
ii
′
is the angle between the
i
and
i
′ axes. Based on Equation 5.6, bulk susceptibility can be calcu-
lated from the HRS emitter hyperpolarizability β
i
′
j
′
k
′
as
=
∑
χ
( )
2
cos
ϕ
cos
ϕ
cos
ϕ
β
(5.7)
ijk
ii
′
jj
′
kk
′
i k j
′
′
′
i
′
j k
′
′
where the brackets indicate averaging over all the emitters. Considering HRS emitters with a single
preferred axis of hyperpolarizability (as expected for push−pull resonance, see below) and defining the
molecular system of coordinates with the
y
′ axis coinciding with the hyperpolarizability axis, the only
nonzero component of β is β
y
′
y
′
y
′
. As described in detail in the following sections, biologically relevant
SHG-emitting samples are characterized by a distribution of HRS emitters with cylindrical symmetry.
We define the laboratory system of coordinates (
x
,
y
,
z
) with the
y
-axis along the axis of cylindrical sym-
metry. Under the assumption that, within the cylindrical symmetry, the emitters are oriented at a fixed
polar angle ϑ with respect to the symmetry axis (see Figure 5.3a), computation of the tensor components
using Equation 5.7 produces the following nonzero components:
χ
( )
2
=
N
β
cos
3
ϑ
yyy
N
(5.8)
χ
( )
2
=
χ
( )
2
=
χ
( )
2
=
χ
( )
2
=
β
cos
ϑ
sin
2
ϑ
yxx
xxy
yzz
zyy
2
