Biomedical Engineering Reference
In-Depth Information
transformation to express a local frame vector,
v
,
or point in the laboratory frame,
v
′, is performed using
a simple rotation matrix.
v
v
′
′
cos
φ
−
sin
φ
⋅
v
v
x
x
=
(1.4)
sin
φ
cos
φ
y
y
This expression can be rewritten more concisely.
v
′
=
R
( )
φ
⋅
v
(1.5)
Transformation of a tensor is performed using the same basic approach, but with considerably more
elements that must be rotated. If we consider first the most general set of eight nonzero possible elements
within both the laboratory tensor (six of which are unique in SHG) and the molecular tensor when the
z
-axis lies within the
X−Y
plane, the rotation matrix is given by expanding the 2 × 2 rotation matrix in
Equation 1.4 to an 8 × 8 matrix through two successive Kronecker products (i.e., multiplication of each
element of one matrix by another matrix).
(1.6)
χ
= [ ( )
R
φ
⊗
R
( )
φ
⊗
R
( )]
φ
⋅
β
Explicit evaluation of this rotation matrix followed by simplification assuming local uniaxial sym-
metry for the case of SHG results in the following relatively concise 6 × 4 coordinate transformation
matrix.
( )
2
χ
χ
χ
χ
χ
χ
0
2
g
g
i
J HHH
,
( )
2
0
0
f
−
h
−
h
h
β
β
β
β
( )
2
J HHV
,
xyz
( )
2
−
−
−
2
g
i
g
( )
2
J HVV
,
yyz
=
⋅
(1.7)
l
2
0
2
h
f
h
( )
2
( )
J VHH
,
zyy
0
0
i
g
−
g
g
( )
2
( )
2
zzz
,
J VVH
2
h
h
f
( )
2
J VVV
,
L
A concise shorthand notation has been introduced,
9
with each letter representing a different trigono-
metric function of an orientational angle (summarized below).
a
b
c
d
e
f
g
h
=
=
=
=
=
=
=
=
sin
cos
sin
sin cos
cos
sin
sin
φ
φ
φ
φ
2
φ
2
φ
φ
φ
(1.8)
3
2
cos
φ
n cos
cos
si
φ
2
φ
i
=
3
φ
