Biomedical Engineering Reference
In-Depth Information
For SHG microscopy measurements of systems with local uniaxial symmetry with the z -axis defined
as the extraordinary axis and using Lorentz scaling factors in Equation 1.12, only a few of these 27 triple
products are required.
β
β
β
β
L L L
2
ω ω ω
0
0
0
β
β
β
β
xyz
xyz
xx
yy
zz
0
L
2
ω
L L
ω ω
0
0
yyz
yy
yy
zz
yyz
=
(1.14)
0
0
L L L
2
ω ω ω
0
zyy
zz
yy
yy
zyy
0
0
0
L L L
2
ω ω ω
zzz
zz
zz
zz
zzz
l eff
,
l
For a point dipole embedded in an isotropic system, every tensor element is rescaled by the same
constant term. In this limit, the Lorentz corrections influence only the overall intensity but not the
polarization-dependence of the response (i.e., β
= ⋅ ). In the more general case of systems that
are not point dipoles or are embedded in anisotropic media (e.g., fibers), the absence of off-diagonal
elements within the local field factor matrices and L results in the rescaling of each element of the local
β (2) tensor (i.e., β
( )
2
C
β
( )
2
eff
2 ∝ ). As a result, all the equalities between the local tensor elements demanded
by symmetry in uniaxial systems are still present after applying the local field corrections. Only the
relative magnitudes (and possibly phases) of the effective tensor elements may be altered by the correc-
tions. If the scientific objective is to quantitatively relate the laboratory measurements directly back to
molecular-level properties, these corrections should be included. However, for many other objectives,
such as imaging of cellular or tissue structure, they can simply be folded into the working definition of
an effective local tensor and be otherwise neglected.
( )
β
( )
.
ijk eff
ijk
1.3.2 incorporating the influence of Focusing within the
Paraxial Approximation
Before presenting methods for treating the local fields arising under tight focusing, it is useful to first
develop a general framework that is valid in the limit of gentle focusing (i.e., with a numerical aperture
(NA) < ~0.8). Within the paraxial approximation, the electric field for polarized light as a function of axial
position z and radial distance from the optical axis ρ is given by the following expressions, calculated for
a focused Gaussian beam. 5
E
( , )
( , )
ρ
ρ
z
=
e
e
x
x
0
C
ω
( , )
ρ
z
(1.15a)
E
z
y
y
0
W
W z
ρ
2
ρ
2
z
z
C
ω ρ
( , )
z
=
( ) exp
0
exp
i kz
+
k
tan
1
(1.15b)
2
R z
( )
W z
2
( )
0
2
+
z
z
0
R z
( ) =
z
1
(1.15c)
2
+
z
z
(1.15d)
W z W
( ) =
1
0
0
π
W
2
λ
πθ
0
z
=
=
(1.15e)
0
λ
2
 
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